for the small amount of times he has a Jack, your only (reasonable) chance of stacking him is betting the flop.
btw you aren't doing as amazing vs a Jack as you might think. it's def not a complete lock hand or a spot to slowplay.
I can see a check with JJ or even maybe AA (just so you don't have to stack off vs the maniac).
but here i'd just pray he has a J or a straight draw and start building a pot. give him a chance to make a mistake. he can't make a mistake if you check