your turn call was a terrible mistake. While I never condone "probe bets" because I think they generally don't work that well, this is a great spot to bet about 55% of the pot just to see what your opponent does. Your only other option is to check-raise substantially, but this is a bad choice. If you want to be really tricky you could check min-raise, indicating you flopped a set and your attempting to extract value. But if you take that line, you have to be prepared to shove the turn if you get flatted, and the other conditions are that your opponent have to be conservative enough, and more importantly DEEP enough, to make the fold profitable for them.
Basically you sometimes just have to fold kings when an overcard comes. It is honestly a situational thing, the fact that you saw a flop indicates that your opponent has two big cards and is drawing. If you both had made hands, or your opponent had queens, the money might have gone in preflop, at the very least you would have heard about it.