Think yourself lucky it's only a kitten. I had to deal with same problem on our enormous (Long Haired) German Shepherd Dog when he had an upset tum a few months ago! DH got to hold his head while I had to deal with the unpleasant end!
When we got our 2 kittens, the girl was dreadful about covering it up in the litter tray. She would walk away & leave it, and her brother would go straight in & cover it up for her