Yeah, except そんなわけ is 'that kind of reason', not 'what kind of reason'.
In any case, そんなわけ is an expression that is a little skewed from normal use of 訳. そんなわけがあったか = "Is that how it is?" or "Is that what happened?"
そんなわけがあるか = "Is that possible?" or "How could that be possible?"
Your sentence is just a strong version of そんなわけがあるか
The かいな instead of か makes it completely clear that it's a rhetorical question expressing disbelief. かい and な are sentence ending particles on their own so the sense seems clear enough anyway, but the かいな combination does have its own dictionary entry on that exact point.
I see you keep wanting this to be a 'grammar' that you can understand, but honestly, it's just a slightly idiomatic usage of common words and grammar. Noting that かい is a rougher version of か is about as much of a grammar-point as there is here. It's a -very- common idiom though so well worth memorizing.