Preflop
Most of the time K9o is just a fold from SB, but against a min-raise from BTN I guess, we can throw in a call or a light 3-bet.
Flop
A dry K high board tend to favour the preflop raiser, and for that reason we typically want to check our entire range to him. You do lead though, and now he min-click you. This is kind of weird, and we cant fold top pair on the flop getting 5:1. We are also not in stack-off mode with top pair meh kicker, so there is really nothing to do here other than to stick in a call and see, what happens on the turn.
Turn
We dont want to lead again after getting raised on the flop, so checking is standard. If he had bombed the pot, you would have a bit of a decision to make, but he check back.
River
I typically dont like to go for thin value in a situation, where the opponent has played weirdly, and I have no clear idea, whats going on. So I lean towards checking with the intention of calling most river bets from him. You do bet though, and now he min-click you yet again. This is a really bizarre line by him, and it almost looks like, he is clicking buttons and raising you just for the sake of raising. So I am definitely not folding here getting 5:1 yet again.
Results
So he was in fact just clicking buttons on the flop, but unfortunately he then hit some runner-runner to end up with the best hand on the river. That sucks, but for me the more important street was the flop, where I much prefer to check-call rather than leading out.