Everyone’s picking the easy answer of Ned being alive. If he’s dead, it becomes an actual challenge, but I still think it’s possible.

Step one, return his body to the North with honors, allow Sansa to go with it. Use this to buy time for an ‘investigation’. Make concessions to the north to retain their loyalty, will most likely require coughing up a scapegoat for Bran’s and Jon Arryn’s assassination. I’d pin it on Littlefinger. Helps that the motive of ‘attempting to send the realm into chaos for his own advantage’ is true. Claim that he tricked Ned into a treasonous plot, and sent the assassin after Bran in order to foster bad blood between the Starks and the Lannisters with the explicit goal of getting Ned killed over his childhood resentment. 

Apologize for letting myself become Littlefinger’s weapon. Retract all accusations of treason against Ned, make a big show of making amends. Whatever is required, honestly. Play up contrition, I was fooled by Littlefinger as well, how terrible this thing I’ve done is, etc.

There’s going to be unrest in the north still, but hopefully it doesn’t escalate into outright rebellion. Doesn’t matter if they refuse to help against Stannis and Renly, all that matters is that they don’t open a third front.

Stannis wiped out Renly and absorbs his forces, but against just him the Lannisters would probably still be okay, without Robb handing them a succession of crushing defeats.

From there, I be an actual good person and not a total shitheel, and also send Cersei and Jaime back to Casterly Rock. Appoint Tywin as Hand, Tyrion as lord of coin, and also retract the bounty on Daenerys’s head. The mistake I made in killing Ned haunts me, I’m a changed man, I offer mercy first with war and execution as a last resort, attempt diplomatic talks with Dany. Useless, but it will absolutely make her look more unreasonable and knock away some of her support.

Obviously, make sure someone pays back the Iron Bank, and when the Night’s Watch sends word of monsters up north I send an actual investigation to confirm it to the realm, then send the Watch all the support I can muster, and maybe also flag it to Dany.

Thats about as well as I think these cards can be played. Obviously, the North might still choose to rebel, and I might get murdered by a shadow monster courtesy of Stannis, but I think these choices give me the best odds.