I think villain played it poorly on all streets. Check raising the flop with king high doesn't make a whole lot of sense imo because i'm assuming you will never be folding a better hand but will continue with all your hands that are ahead of k hi. I would much rather peel in villains spot and make a decision on the turn based on my opponents barreling tendencies, opting to usually call down unless my opponent will often shut down after the flop c-bet if I peel on a dry board such as this. I would rather check raise with a hand that can't profitably peel. With a hand like jt (which actually does have some showdown value), I would still just check call the flop, and possibly the turn too if villain is known to double barrel light but often gives up on the river.
Spots like this are interesting, and imo the correct way to play them depends on how you play other hands in similar spots. If you are 3-betting most of your aces preflop, you don't have a lot of room to make bluffs on boards like this, since there aren't a lot of hands in your range that you can check raise for value. For my playing style, I call with a lot of my aces preflop and will check raise a flop like this with any ace or queen. This gives me the opportunity to throw in more bluffs with hands that completely whiff, since I do have a lot of value hands that won't be folding.
All of this is from villains perspective, the way you played the JJ was completely standard.
But what do I know, I never get to play this game anymore.