No experience with either of those. But I have heard recently people were having trouble getting Colt to respond on some customer service.
Call me a curmudgeon, and no disrespect intended toward this poster, but the OP asked for folks who'd had experience with either or both, not anecdotal, third-hand, "I have heard ..." responses.
I happen to own two Cobras and one King Cobra. In my estimation, the triggers cannot possibly be matched by any current production revolver at the price point. They are solid, reliable, accurate, and did I mention the triggers are superb?
I have shot the original Kimber K6 extensively. I do want one, just have had other priorities for now. I found the specimens I shot to be accurate, easy shooting (although I'd prefer slightly more hand-filling and tacky grips over the wood). I liked the sights. The trigger was okay, but I am not a fan of DAO (which is why I waited for the DA/SA version) but not great. It's a nice package, they're some very pretty revolvers coming out now with great fit and finish.
A poster stated the Kimber has a greater choice of sights; this isn't true -- I don't think this member has seen the last couple models of the Colts that were released. You've got fiber optic, night sights, and brass bead front sights and a target model.
Kimber has that "Royal" version that is sure pretty, but as far as finish? Isn't this for use as a carry gun? The Colts I own are nicely finished and the fit is excellent. But, if cosmetics are that important, a couple of the Kimber models may have the Cobra beat in that regard.