I like the sizing preflop - then the flop rolls out and I would check/call for sure. When we bet this flop we are only getting called by sticky underpairs (I'd say TT and lower), flush draws, and Ax - we do not hold the K of hearts so we do not block any flush draws the OP may have and as such this means the nit flush draw is entirely possible to be in their range.
Anyhow, check flop because we are generally only getting called by better and it under reps the strength of our hand. If our opponent bets, we call - if they do not bet and they check back - then now I am going to be betting the turn ESPECIALLY when the heart comes. Why? Because our opponent should nearly ALWAYS be betting their nut flush draw on the flop when we check to them and offer up the betting lead. So for me, this rules out most good flush draws - so we bet small, 33% - 40% pot and if we get raised this is an easy fold if we do not get raised we can evaluate rivers accordingly.
If we were to bet turn small and get called and then the river comes the heart - now we can stick out a bigger bet, somewhere around 2/3 pot. Looks suuuuper valuey and is not easily called down by a smaller heart. I think every heart T and under folds. You are potentially folding out the J pretty easily as well, it depends on our opponent.
As played I think the flop bet is a mistake and we are generally getting called by better - turn card doesnt help us much at all - so we check, they check back. River is another heart - I dont think I would bet here too often, my hand still has showdown value and most players aren't into turning their hands into bluffs on 4 liner rivers like these so I'd check and hope he checks it back, fold to any bets and move on. You cant win em all and it's not the best runout for KK.
For me villan made a mistake in this hand as well - when you cbet flop and check back turn - I know you NEVER have an Ace or a big heart - so when you bet river I know you have some bluffs and smaller hearts in your range so I may do something like raise you 2.5x (small and value looking) and make your life miserable