If you've seen some of my other posts and threads, you'll know I'm one of those people who insist that KK is quite foldable preflop, given the right circumstances. Unfortunately, those circumstances essentially never exist in anonymous games. I'm afraid you're stuck playing this one out.
@DrStrange hit the nail on the head. Can't fold, and a raise would be a bad idea because basically only AA will ever call, so just flat the preflop raise.
And then this flop. Meh. The check-check is interesting. Hard to see any overpair checking back that flop, and that means AA as well as QQ. Between the 4-bet and the check back, I suspect AK, the only arguably 4-bet-worthy hand that would want a free card on this flop.
The turn makes the board look very threatening, but really, what hands can Villain reasonably have that hit the straight? Really just QQ unless he's an especially loose 4-bettor. But in this spot, with the pot that big and the board that wet, who's checking back with QQ? Not that it's impossible for this to happen, but the information we have indicates it's improbable.
I don't think you want to check to Villain on the turn and give him an opportunity to take a stab that you can't counter without shoving. I think the best move here is to put
him in that spot with a modest but decent-sized bet, like you have a Q and want some easy action. Say, $9.75.
The outcome should be roughly binary: if Villain has a Q (likely QQ), he'll probably raise. If not, he'll probably fold. Either way, you get a lot more information from how he reacts to this bet than you would if you were to check and he were to bet. If he just calls, that's not good either, and you should probably not invest any more in this hand.