I usually don't raise K6offsuit (even with position), unless I'm up against a bunch of nits. If you hit your K, you probably will only get paid off by a bigger K (lots of people will call with K10,KJ,KQ otherwise you basically have a bluff.).
I usually give up on the turn. When villian called your raise with on the flop, he obviously likes his hand. Nothing has changed on the turn, so I don't see this villain folding here.
I'm not a pro, but I'd like to discuss. My play/thougthprocess would be someting like this:
Since villain is shortstacked and we have a drawing hand I'd bet smaller on the flop.
If I'm still drawing on the turn I'd check behind and give up if villain bets river.
Id like to play it fast. Since he is short stacked and against any peace he has got from this board or any overcards we have good enough hand to play for 22bb + we have fold eq if we bet