For someone who's most famous piece of advice is to never let others know what you want, telling Jon outright during their first meeting seems a bit reckless. On the surface, his motivation for doing so seems clear enough. We know that he wants to marry Sansa for political and/or personal reasons, and obtaining The King in The North's endorsement for the union would likely increase Sansa's chances of compliance and the legitimacy of the marriage itself in the eyes of others. Apparently, he would obtain that blessing in exchange for saving Jon from being flayed alive at the hands of the Boltons. Fair enough, right?
But LF isn't that naïve. He had to have known that there was a chance that Jon would be offended by the idea, therefore souring relations with him and his chances at a marriage with Sansa. Was it simply a risk he was willing to take? I doubt that, as it seems way to chancy an endeavor for LF's fashion. Although he might not have expected a death threat out of the deal, the situation was really a win for him either way. And by threatening him with death, Jon has given LF a brilliant opportunity to play his cards well, which I will explain later.
Back to the present. Apparently out of lack of either concern for his head or faith in Jon's return, LF goes right back to talking to Sansa after Jon leaves, a direct violation of Jon's threat. And he's doing nothing to hide it either. I also doubt that he considers Jon's threats to be idle. Rather, he is getting ready to play his cards, and he's got an opportunity to do it well. His plan is as follows:
But why would Sansa strain her relationship with Jon to defend LF? Simple: The Knights of the Vale. Yes, they are currently aligned with Jon, but that's only because Sansa is as well. LF will take the KotV wherever Sansa leads, but if LF was to die, the KotV would likely merge with Jon's north. Essentially, that would be Sansa's final surrender of any claim, power, or influence in the North. And, if LF's "seeds of discontent" have taken root, she won't be willing to do that. Sansa has seen the value of the KotV. And she knows that, if push comes to shove, she's going to need them as leverage with Jon. She has to bring something to the table, she's learned that her name wasn't enough. Otherwise, she might as well be any other lady.Jon returns from Dragonstone and quickly learns that LF has been talking to Sansa. He begins to arrange for his execution but Sansa, either at her own will or with LF's encouragement, begs Jon not to do so.
Side note, I wonder how the nearly inevitable meeting between Arya and LF will go. Although she might doesn't know it, Arya owes him big time for him going against Queen's orders and not turning her in to the Lannisters when they met at Harrenhal.One of two things will happen.
If Jon agrees to spare LF, he will keep his head and enjoy even more strained relations between Jon and Sansa to use to his advantage.
If Jon refuses to spare LF, Sansa is left with a choice: Jon or LF. If she chooses Jon, LF is as good as dead. But, she will be surrendering all her power and influence, as previously mentioned. But, if she chooses LF and works out a few technical details like sneaking the both of them out of Winterfell, she goes rogue and has a giant army at her back, presumably to use to set up her own claim to the North. LF is apparently banking heavily on the second option. Why? Because, for one, he keeps his head. And secondly, Sansa will be completely dependent on him. He could then extort her into marriage if that's what he wants. After all, he is the link between her and the KotV. Should Sansa be given that ultimatum, LF is hoping that he has driven the wedge between her and Jon deep enough that she will choose his side.