When you check the flop (by mistake) your hand is obviously under repped and he maybe thinks his pair of jacks is good on that runout.
With the gutshots and flush draw it is a dangerous board for Kx to give free card on.
I would probably expect Kx to bet for value and protection and once you checked it looks like he discounted top pair+.
The river bet does look a bit thin, you won't have many worse jacks and a midpair (88 99 TT) probably doesn't pay off on this sizing. If he was bluffing
the only hands you might lay down that he loses to was what, QQ AJ, or a chop?