Write to ExpertFlyer and tell them you want to see KE's award availability (other than first class, of course)
I once did it, and got a reply saying "We'd love to do so however it is not up to us, KE has to make the inventory available for us to see which they don't.". I don't believe it, because (1) EF shows KE's availability in F. I can't think of a reason KE releases F award availability but not J or Y. (2) NWA.com used to show KE awards, so there should be a way to do that.