The only "stuffed animal hospital" I ever used was called MOM. Whenever my toys or my nieces stuffed animals got so bad they were breeding need brands of bacteria, my mom, (or in my nieces case me) would toss them in the washer with soap, some bleach and a touch of vinegar. She'd then take the hand dryer to them, fluff the animal up with a brush, and I'd have a brad new toy!
When the animal really got to be beyond repair, we'd put the toy in a special bag for the special "Animal Maker" to remake (mom threw it away when I wasn't looking) and she took me out to choose a new specie stuffed animal, usually under $5.
Anyway, my point is that you might want to try cleaning the toy yourself before you send it off to a professional place, because I'm sure they're quite expensive. You might want to ask a dry cleaners if they will do the deed for you. Sure, they do clothes, but they might be willing to do the toy,especially if you are on good terms with them.
Good luck!