You have to be slightly worried betting pot on turn and getting called right? He could easily have Ax, any pocket pair, and suited connectors(56, 67, 78) preflop. Over pairs are unlikely as he would have probably three bet pre with 10's+. I just question what he calls a shove off on the river that doesn't beat you, maybe A8 (unlikely though).
I'd 3x pre.
Flop looks good.
Turn seems a little bit too big, but I dunno how stationy the average 10nl reg is so ehhh.
River bet is too ambitious, especially given sizing - I'd imagine it's gonna be tough to get called by a worse one pair hand here with your sizing (or maybe even get called for any sizing OTR). Definitely check or bet smaller OTR IMO. Given your turn sizing i guess check back OTR
If you have no intention of folding your KK you might be thinking about shoving the turn. It is a bit big but you will prevent your opponent from drawing out. You will be running in flopped sets eventually though.
Otherwise it is better to bet/fold the turn with around 60% of the pot.
As someone said, villain flopped double gut shot on flop. He then picked up a pair in the turn, so probably not terrible to call the turn given the equity and assumption you'll stack on the river.