Well, you have to put your opponent on a hand that beats you which currently is only Kx. Think about his range of hands, what type of player is he? What types of hands would he continue with.
Some important things to consider in factoring in a decision are: What was the preflop action? If he's a thinking player who is positionally aware, what position is he in preflop? Also, what are the bets and his stack size relative to the pot?
If his range is very tight we can assume he's playing only strong Kings. For example, AK, KK, KQ, KJ, maybe K10 but that could be stretching it. We can assume that he would re-raise, or 3bet KK and maybe AK so that might leave you with just KQ/KJ/maybeK10 in his range. Again, I don't know exactly what his range might be, I'm just going off of the fact that much info is given.
Think also about, how he'd play those hands too. Would he slowplay KJ on that flop? Does everything make sense up to this point?
If you think he has air a majority of the time, plus factor in your 10 outs to hit a better hand, it might be profitable to continue.