I think we should bet something on the river. I feel like AA would bet the river after you check back the turn, so he could have queens or maybe he 4 bet a hand like KJs and will pay us off. a 1/3rd potish bet while looking kinda like what it is, I still think just gets a sigh call from those hands that we beat.

Given that he has 100% 3bet so far and now 4 bets you makes me think that he is empty 90% of the time when he checks both turn and river. I would bet something like 13$ on the river to get him a good price to bluff shove or maybe tilt call with smaller pockets, A high or a jack. 

His play makes sense if he has a 10 tho because your hand is a little bit disguised and when you check turn it looks more like you are on a draw than a pair. So lets say that you had KQ, that would probably be a shove on the river if he checks to you?

Anyway, i think kings are to strong to check river with!

I think you can easily go for 2 streets of value in this spot. 3 streets is beginning to be a little too much, since not much is going to call and be worse than us, but 2 streets is fine.

Since you have position, I like the check back on the turn, as this gives the villain an opportunity to bluff the river. Plus, if he doesn't, you can still bet the river yourself.

Betting the turn would not be bad as well, but I would prefer to check the turn in order to extract maximum value out of his bluffs. 

If it is checked to you on the river, the villain has made it clear he will not bluff at the pot, so he either has an absolute monster (not very likely, given the way he played the hand), a value hand that is worse than yours, or nothing, and decided not to bluff. 

Because of that, I think Robin's idea of betting smaller in order to induce a bluff is out of line. He's not going to be bluffing in this river often at all, since he showed no signs of aggression when given the opportunity twice.

I would bet  just enough to make him call with AJ/ish, Queens, etc, and fold if he raised, since we're going to be beat most of the time (if he raises that is).

Good luck on the felt, 

nH.