If your opponent is Buckster, easy fold after river.
Otherwise, i'd slow-play and kick myself hard when last K flops and opponent flips the 4th K. That hand, you're going to lose to royal flush and 4 of a kind (K's) and beat everything else. I don't know how big opponents stack is, but i've got 15,000 more chips to bet, and I want as many of opponents as possible. I'd check, and hope opponent hits flush. Then check again after turn - i've got 35,000 chips at the moment, but want 50,000, and if i'm up against a 1 outer, i'll take that risk gladly. If you push, he'll fold vast majority of the time - i'll let him try and get straight or flush.
But as I said, against Buckster i'd just fold