Again, you might think i'm sounding incredibly nitty, but 1st level i'd even fold this. Pair of kings is nothing in O/8, you've not got a playable low, no nut flush draw.
as played, the post flop bet is fine - you want to win it there. After that, you need to slow down, you may have the nuts now, but any diamond and possibly any 9, 10, Q or A then you're behind for high. Any A, 2, 4, 5, 7 or 8 then you're splitting (and will have to stand a big bet from someone with a made low and a flush/straight freeroll for high). all you've got for you are one King, 3 6's 3 3's and 3 jacks. Check/fold the turn.
On river, if it was just winerall maybe i'd call and hope he had busted draws, but it's just a hopefull call that neither has got 10, Q. I think you've bet yourself into a must call, but I personally wouldn't be in that situation here.