Eddard has his first council meeting and learns that the treasury is bankrupt due to Robert's lavish spending, presumably on fatty foods. Afterwards, Littlefinger leads him to a secret rendezvous with Catelyn. Thinking Littlefinger was just stringing him along, Eddard almost punches him, but Ser Rod sans barf beard appears to confirm that Cat was there. Did Eddard really think Littlefinger took him to a secret passage that led outside the castle, then down a rock cliff, then on horseback to a brothel just to set up a one liner "your wife is inside" joke?
Littlefinger delivers zinger after zinger, taking nothing seriously apart from his feelings for Catelyn. He shrugs off the huge royal debt, suggests that Eddard fondle some breasts, and sarcastically golf claps when Eddard first recognizes Catelyn. I was disappointed that Littlefinger didn't ask about Jon, and that Eddard didn't politely request some time alone with Catelyn. Come on dude, it's been a while since you've last seen your wife, and you're already in a brothel. Eddard should have done it just to see look on LF's face.
After hearing Catelyn's story about the assassin and how Bran's wolf saved both their lives, Eddard suddenly realizes that he probably shouldn't have killed Sansa’s wolf. Littlefinger suggests that they just pretend the super recognizable Tyrion knife doesn't exist, but Eddard is too proud and too righteous to forget about something of this magnitude, even if it’s in his own best interest. The Lannisters had enough power and influence in King’s Landing to murder the previous last Hand, so Ned will ultimately hang his hopes on the King. However, Robert is politically inept and disinterested, allowing his council to run the kingdom while he requests another party.
The Night’s Watch is such a raw deal. You can’t own land, you can't have sex, your brothers are the dregs of society, and the guy training you is Alliser Thorne, an anal, humorless, tool of a man. Predictably, Tyrion mocks him so badly that Thorne is forced to leave the dining hall, much to the delight of the others. After dinner, the Lord Commander pleads with Tyrion to convince the King to send more men. Mormont continues on about abominable snow men sightings, missing uncles, and mountain people fleeing, but Tyrion doesn’t take him seriously. Maintaining the Wall is like paying for asteroid insurance -- it guards against a disaster that happens so infrequently that it’s easy to forget about it or dismiss it. Robert isn’t going to send men and resources when he can't even pay his own debts.
Before Tyrion departs, Jon gives him a few messages to deliver to Robb, Rickon, and Bran. Didn’t Tyrion recently try to whack Bran? It would be supremely stupid to return to Winterfell. Either Tyrion thinks the Starks don’t know it was him, or Jaime and Cersei acted without his knowledge.