I cant remember much about villain this was almost a week ago and I have no HUD, I think he was your average player.

PokerStars, Hold'em No Limit - $0.01/$0.02 - 9 players
Replay this hand on Upswing Poker

UTG: $1.09 (55 bb)
UTG+1: $1.94 (97 bb)
MP (Hero): $0.98 (49 bb)
MP+1: $1.46 (73 bb)
LP: $1.32 (66 bb)
CO: $1.56 (78 bb)
BU: $2.77 (139 bb)
SB: $2.35 (118 bb)
BB: $1.98 (99 bb)

Pre-Flop: ($0.03) Hero is MP with K♥ K♠
UTG calls $0.02, UTG+1 calls $0.02, Hero raises to $0.10, 1 fold, LP calls $0.10, 6 players fold

Flop: ($0.27) A♥ 9♠ 5♠ (2 players)
Hero bets $0.10, LP calls $0.10

Turn: ($0.47) T♣ (2 players)
Hero checks, LP bets $0.10, Hero calls $0.10

River: ($0.67) A♠ (2 players)
Hero checks, LP checks

with the ace and flush out there no i dont value bet the river. If he were checking to you then maybe but i still would fear the flush. with him calling all the way i figure he is looking for the flush draw

Getting to showdown with your KK out of position on that board is a good result. I think checking is fine.