If Brienne is, as many suspect, taking Jaime to Lady Stoneheart (LS) for judgement, how do you WANT things to turn out? How do you think things WILL turn out?
Personally I am hoping that Brienne delivers Jaime to LS (Jaime doing an incredulous dissapointed-face at Brienne when he realises) and a trial by combat ensues. If this is the case I'm willing to bet my right hand (get it?) that Stoneheart will nominate Brienne as her champion to fight Jaime. If this happens, I'm thinking Brienne will have a serious crisis of loyalty, go into full on chivalric meltdown and refuse to fight or let Jaime win, at which point he will refuse to kill her and they will end up attacking / fighting the Brotherhood together.
Being a massive Jaime fan I'm of course hoping for a last man standing type deal where he emerges from the fight drenched head to toe in blood, with Brienne essentially sacrificing herself for Jaime. With, LS, Thoros, dead (with no more 1up mushrooms) and whatever's left of the Brotherhood dead / run away, maybe GRRM will even break convention and have a few last words from Brienne, after which Jaime is fully catalysed into becoming a proper knight once and for all.