Re: Is this play with KK ok?
 

Value bet river here especially in nanolimits unless specific read says otherwise.

Re: Is this play with KK ok?
 

Beautiful. The only thing I can see being changed is to value bet the river. He capped PF so a big pair is likely. Judging by his passivity, it isn't aces. He could have AK, but that's his only reasonable holding at this point that you are behind.

The check-through is fine and I'd do the same, but I thought I'd toss the idea of a river value bet out there.