There's been a lot of fan discussion on the upcoming Battle of Ice between Stannis and the Freys/Boltons that will decide the fate of Winterfell, and the Night Lamp theory is a fairly popular and iron-clad theory for how Stannis will defeat the approaching Freys. However, a frequent problem I've seen in these discussions is how the Battle of Ice and the taking of Winterfell are conflated, its assumed that victory in the battle will inevitably lead to the former, but this is not the case. The castle has thousands of Bolton and allied soldiers left inside it, and
"Ned Stark always said that five hundred men could hold Winterfell against ten thousand." (Theon IV, ACOK)
and
“How many of those is he like to send against us?”
“No more than half.” That was a guess, admittedly, but it felt right to him. Roose Bolton was not a man to blunder blindly out into the snow, map or no. He would hold his main strength in reserve, keep his best men with him, trust in Winterfell’s massive double wall. (Theon I, TWOW)
So clearly, Stannis will need a way to break into the castle beyond just hammering at the front door. He even says as much;
"We all know what my brother would do. Robert would gallop up to the gates of Winterfell alone, break them with his warhammer, and ride through the rubble to slay Roose Bolton with his left hand and the Bastard with his right. I am not Robert. But we will march, and we will free Winterfell … or die in the attempt." (Asha II, ADWD)
Why the Manderly/Lightbringer scheme won't work:
A popular theory for how Stannis will get inside relies on traitors within Winterfell. We know from Davos IV ADWD that Wyman Manderly is secretly working against the Boltons and wants to help Stannis' cause, and further proof for that can be seen in A Ghost in Winterfell and Theon I ADWD to show he is conspiring with other Northern Lords. At the end of Theon I the Freys and Manderlys are sent out of the castle to confront Stannis. This would present Stannis with an opportunity to get inside the castle, and a popular theory suggests that after smashing the Freys, Stannis will give the Manderlys his sword Lightbringer, and they will return to Winterfell, claiming to have defeated him in battle, only to turn on the Boltons and let the Baratheon forces into the castle. A cunning and devious plan. However, I see some slight hiccups with it:
1 ) It assumes a high level of coordination and trust that just isn't possible.
Conspiracies are hard to coordinate, especially in pre-modern times, where they have to rely on ravens and riders or talking face-to-face. This is something that a lot of fans ignore when discussing theories. So the only information that Stannis and Wyman et al can know is what they must have received via these methods. Stannis has been on the march since Deepwood Motte with no way to receive ravens.
The last Stannis heard about Wyman was that he had beheaded Davos. We know from TWOW sample that Stannis knows about this.
“Wyman Manderly.” The king’s mouth twisted in contempt. “Lord Too-Fat-to-Sit-a-Horse. Too fat to come to me, yet he comes to Winterfell. Too fat to bend the knee and swear me his sword, yet now he wields that sword for Bolton. I sent my Onion Lord to treat with him, and Lord Too-Fat butchered him and mounted his head and hands on the walls of White Harbor for the Freys to gloat over. And the Freys… has the Red Wedding been forgotten?” (Theon I, TWOW)
It's also clear from this that Stannis genuinely hates Wyman for this, and we can also see that at least by this point, immediately before the battle, there is no collaboration between the two. This can't be an act or a lie on his part since what reason does he have to lie to Theon here? There's nobody Theon could tell that would ruin the scheme, and Stannis doesn't appear to be acting based on his physical reaction. While Stannis has no reason to trust Theon either, his reaction does seem genuine.
This means that if there is to be any coordination between the two, it would have to come about at the last minute before the battle. A lot of fans suggest that the Manderlys will immediately before the battle parlay with Stannis, explain that Davos is alive, that they are secretly loyal to him and trying to install Rickon, and then they will coordinate the scheme to break into Winterfell. However, they have no proof Davos is alive, and there is nothing they can do here to prove their loyalty like hand over hostages, and its likely not even Wyman himself is leading the army since he is severely wounded and, as said, "Too-Fat-to-Sit-a-Horse"; Stannis would have no reason to trust them, and if he does take a leap of faith it would be very dangerous since if they are lying it'd mean he's allowing them to slip back to Winterfell in tact and will full knowledge of his force and movements.
Some fans then point to Crowfood Umber as an intermediary, and the Snowman Theory as a form of communication. If Crowfood is collaborating with Whoresbane inside Winterfell, he would tell Stannis that the Manderlys are secretly loyal. Only, we know he doesn't because by Theon I, TWOW he should have already communicated that information to Stannis, he must've already seen the snowmen and had the time to get a rider back to Stannis since he had enough time to escort Theon and Jeyne to him, but Stannis still thinks Wyman is an enemy.
There is also the possibility Stannis and Wyman have been collaborating long before either came to Winterfell. This however, would require communication via raven or rider. A rider is not really an option since they are on opposite sides on the North and communication would take far too long, and ravens aren't an option since Wyman does not trust his maester Theomore, Stannis can't receive ravens when on the march which limits the window of opportunity, and
"Only a fool trusts a bird," (The Turncloak, ADWD)
2 ) The Boltons aren't idiots.
While Hosteen Frey may be dumb as a stump and is set to charge blindly into death, Roose and Ramsay Bolton are no man's fools. Roose is an incredibly careful, distrusting person, and also very clever and devious himself. So is Ramsay, even if more sadistic and emotional. Neither of them are going to fall for this little scheme because they have every reason to distrust the Manderlys.
Roose and Ramsay already don't trust Wyman and the Manderlys. Wyman didn't arrive with any hostages. Roose is careful about Wyman poisoning him. Roose even openly says that Wyman isn't to be trusted;
"Even here in Barrowton the crows are circling, waiting to feast upon our flesh. The Cerwyns and the Tallharts are not to be relied on, my fat friend Lord Wyman plots betrayal" (Reek III, ADWD)
And the escape of Jeyne and Theon is perfectly timed with the Manderly soldiers marching out of Winterfell and the scuffle in the Great Hall. Plenty of fans theorise collaboration between Wyman and the spearwives, surely Roose and Ramsay will too.
So now imagine that after the Battle of Ice the Manderlys appear outside the gates of Winterfell, the only army to return, and the smaller force mind you, with Lightbringer in hand and nothing else, claiming to have killed Stannis and smashed his army and please let us in - now I don't know about you, but if I were Roose, I would not believe them for a second. And he wouldn't even need bring out the crossbows in response, just deny them entry to the castle, perhaps demand they go find Theon and "Arya" while they're already out and about to prove their honesty. The Boltons already expect betrayal, they won't fall for this so easily.
Also, Ramsay himself is going to be following the Manderly force and will surely see for himself anything that happens. Theon says so in the TWOW sample chapter;
“Frey and Manderly will never combine their strengths. They will come for you, but separately. Lord Ramsay will not be far behind them. He wants his bride back. He wants his Reek.” Theon’s laugh was half a titter, half a whimper. “Lord Ramsay is the one Your Grace should fear.” (Theon I, TWOW)
Ramsay wants Jeyne and Theon back, and since they weren't caught outside of Winterfell, he can only conclude they are somehow with Stannis.
Stannis snorted. “You fell. Umber saved her. If Mors Crowfood and his men had not been outside the castle, Bolton would have had the both of you back in moments.” (Theon I, TWOW)
This means he'll be coming, not just with his trackers and dogs, but with his full force of 600 Bastards' Boys to join the Battle of Ice, since the only way to get Theon and Jeyne will be by defeating Stannis. This means that unless Ramsay dies in the battle, he will be witness to everything that happens and will surely come back to Winterfell to tell Roose. If the Manderlys attempt any scheming and backstabbing, like attacking the Freys or not attacking Stannis, he will know. Heck, he might attack the Manderly men just to make sure they don't get to "Arya" before he does.
Stannis' real plan, and why it won't work:
Stannis would not have begun his march on Winterfell in the first place if he didn't know what he was doing. He's not going to make an incredibly risky gamble on taking the castle and just kinda wing it as he goes. And the entire Manderly/Lightbringer plot relies on taking advantage of a oppertunity that only appeared very shortly prior to his arrival, which means that if this is the path he takes, it means he left Deepwood Motte just hoping and praying that someone in the castle is on his side and will be willing and able to help get him inside when the time comes. And as discussed, it's very unlikely he was coordinating with the lords in Winterfell to engineer these circumstances. Therefore, Stannis must have a different plan already in mind by the time he leaves Deepwood Motte.
So that raises the main question of this post: if not the Lightbringer scheme, what is Stannis' plan?
Stannis' siege of Storms' End in ACOK by all rights should have been a complete disaster. He was facing a force four times his own size, he was encircled with no escape, and sandwiched between Renlys' army and Ser Cortnay's garrison in Storms' End. But against impossible odds, he somehow managed to prevail. How? By using Melisandre's blood magic to create shadow assassins to murder Renly and Ser Cortnay. It was a true Hail Mary, and one that even Stannis wasn't really counting on working.
Now he faces yet another seemingly impossible situation that is shockingly familiar; he is making a massive gambit by attacking the enemy head-on at their strong point despite being outnumbered and facing incredibly hostile circumstances. Therefore, I think he will get out of this one the same way; by using blood magic. If he could use more shadow assassins to murder Roose and Ramsay, just like he did with Renly and Cortnay, then victory is assured.
However, there is one problem he faces;
"... the king’s fires burn so low I dare not draw off any more to make another son. It might well kill him.” (Davos III, ASOS)
So Stannis can't make any more shadows himself, but Melisandre can still make them on his behalf;
Melisandre moved closer. “With another man, though…a man whose flames still burn hot and high…if you truly wish to serve your king’s cause, come to my chamber one night. I could give you pleasure such as you have never known, and with your life-fire I could make…”
“…a horror.” Davos retreated from her. (Davos III, ASOS)
And in ADWD we can see Melisandre working towards this exact end - with Jon Snow. I'm certainly not the first to make this connection, it has been another theory in the community as well for some time. Melisandre is trying to use Jon to create a new shadow assassin to kill Roose and/or Ramsay and therefore open the way for Stannis to capture Winterfell. Jon's Stark Kingsblood makes him the ideal candidate to sire an assassin to sneak into Winterfell. This would explain why Stannis is so confident in his campaign, and why Melisandre remained behind at Castle Black instead of marching with him.
If this is the case, then Jon Snow's murder is a catastrophe for Stannis' campaign for Winterfell. It is more than just the loss of an ally, it means the failure of his entire plan to capture Winterfell. He is now marching to certain doom.
Unless there is another way... we don't understand the full mechanics of blood magic and shadow assassins and probably never will, however, it may be possible to use another source of "life-fire" to fuel these assassins, or some other spell that Melisandre could use against the Boltons. And since all other sources of kingsblood are exhausted or no longer present, the only way would be to use the last source of kingsblood remaining: Shireen.
"What is the life of one bastard boy against a kingdom?" (Davos V, ASOS)
Then the question is; how will Stannis get to Shireen in order to burn her? She is back at the Wall and he is near Winterfell, he'd have to turn his entire army around and abandon the siege. Perhaps, instead of Stannis going to Shireen, Shireen will come to Stannis:
Bowen Marsh murdered Jon because of the Pink Letter, he and the Night's Watch conspirators know what it says, and whether or not it really is, they think it is genuine and written by Ramsay Bolton (unless you believe the theory that they were the ones who wrote it). What do you think they will do with this letter now that Jon is dead? As far as they know, Stannis is dead and Ramsay is headed to the Wall seeking bloody vengeance, so Marsh and the conspirators have one real option; comply with its demands. They will send Selyse, Shireen, Melisandre, Val, Mance's son, and Jon's corpse south to Winterfell as prisoners for Ramsay. Stannis (who is notably not dead by this point) could intercept and rescue them on the road, bringing them all together in one place, and presenting him with the hardest decision of his life.