How to diagnose the problem?

As a cat owner, I feel your pain. I swear the darn things sit around while we are at work and plot new and interesting ways of screwing up our lives.

This one should be pretty straight forward to fix. It sounds like he hurled right into the fan and that probably made its way onto the board. Tear it down, remove the heat sink and fan and clean everything out. Use copious amounts of 99% IPA on everything, including the fan. Use extra on anything that will get hot because burned cat puke can't be a pleasant smell. Hopefully a good cleaning is all you need.

I can't imagine this killing anything besides the fan, which is a pretty simple and cheap fix.

Teardown for your reading: Xbox One Teardown