Preflop
Not much to say here. Obviously we always open KK, and 2,5BB is good sizing.
Flop
As SpanRmonka say, your rational for betting this flop is flawed. We never bet "for information" or "to find out where we are". We bet for value (to get called by worse hands), as a bluff (to get better hands to fold) or for
equity denial. On this board you will never get a better hand to fold, so you are not
bluffing. And if you are not happy, when your opponent continue and make a small bet on a completely brick turn, then you are not really betting for value either. Finally your hand dont need much protection, since the only possible overcard is already on the board.
And since you have no good reason to bet, you should check. Checking does not mean, you are giving up, because this hand can easily check-call. And this is something, we should be doing a lot out of position, even when we were the preflop raiser. So if you are not comfortable taking these kind of more passive lines, then you need to get used to it. Solvers C-bet way less out of position than in position. And so should you, even if you are playing in the micros.
You might also need to think about this spot a bit differently. Your current thought proces is likely something like "oh man, there is always an ace on the flop, when I have KK, why am I so unlucky". But instead you want to think of this situation as "I am out of position with second pair and a gutshot". Because then rather than having an emotional reaction to a bad flop (which of course it is), you are accepting the situation and focusing on making the best decisions.
And the truth is, we are not always going to win the hand or even get to showdown, when we are out of position with second pair and a gutshot. So part of the mental game is to accept this and get rid of entitlement tilt, just because you had a very strong hand preflop. At this point there is no real difference between having KK and KQ, so there is no reason to think about those situations differently.
Turn
As played this is a check-call, when he only bet 35% pot. You will of course be behind here quite often, but you are not drawing dead, unless he has specifically KT, which you block. Against everything else a K or T will often give you the winning hand or at least a chop. And sometimes you have the best hand already. He is not only floating you on the flop with top pair or better. He could have a flushdraw. Or he could have a worse made hand, that he should be checking back, but he bet it anyway. Like KQ or QT.
