KK river sizing logic check

It's a tricky one, betting is definitely superior to checking.

I think betting $5 is good because he might call with TT-QQ thinking we are bluffing the ace, also his range is weighted more towards PP's it's a lot more likely he got to the river with TT-QQ that AJ-AK. Barring AJs-AKs but he may raise the flop some % of the time as well and try and get stacks in with these hands so its slightly less likely again he has one of the few Ax combos and he probably doesn't call pre with ahnds like A3 A6 to a 3bet from the blinds.

i like your line a lot
as jkds said, there is some risk of inducing a bluff on the river with your sizing, but it's not likely at all at 5nl given that aces are scary for most of their range and they're not going to be the type of player to bluff JJ into a river ace