My guys normally got gunky eyes when they had an URI. If it got really bad, and the antibiotic didn't do much for it, I use a eye med from the vet. It's like a thick salve. I'd definitely check with the vet to see if it's ok to use on a young kitten. We had one at the shelter that got really bad, and we thought she was going to lose her eye. She did get better, thank god. She is such a sweet and gorgeous little girl (she's also Feline Leukemia positive, so her immune system is not great... which probably added to the eye getting so bad).
The warm washcloths area good idea. I use those on my guys when their eyes are gunky.