AKo

You beat a number of K-high combos. Say KQ, KT, K9, K8 + K7s, K6s, K5s, K4s, K2s, that could reasonably play this way. This is 68 combos. Throw in a few straight draws + A4s/A3s.

There are 13 sets and (reasonably) 12-14 two pair combos (14 combos would include 43s).

Even if he plays fewer kings this way, you are probably solidly ahead.

I'd be more concerned on a K♥Q♠9♣ board. Now your range has hit the board very well.

About capping OOP to take initiative: Let reads and the board decide. As I default I enter calling mode. It sucks when they check back on turn, but they have still payed 3 SB to draw instead of 0 SB (by checking the flop), so they have already paid for their free card. It becomes pretty expensive for us if they raise on turn if we get greedy with the wrong read.

turn isn't overplay. the hand is wp you cant really do anything else here.

There is nothing much too add than saying well played. This time he did prolly have a set or KJ but as yohann mentioned correctly, he does have too many worse Kx and maybe pocketpairs that he's still calling down. On a more scary Board as KQ9 it's just a calldown if you're being checkraised. K9 KQ and especially JT are too many combos that have you beat. If he'd raise IP and you are OOP, you should just 3bet right away and fastplay your hand.