I think you have to shut down once that K comes. Even though it made you two-pair, it's THE worst card that could have fallen for you (with the possible exception of the J or T pairing). With the board one face card off a straight, it is simply a bad play to give action unless you have a read on your opponent (namely that he will frequently bluff scare card situations like these). I almost definitely wouldn't bet, because the only way you're going to win any more after that K comes is if he has you beat or has an inferior two-pair and hopes you're bluffing the straight. If I was in this situation, I would check on the turn and probably fold if he makes a sizeable (>1/2 pot size) bet. If he checks behind you, I'd check again on the river and again fold if he makes a good-sized bet.