There is a little bit of debate about whether shaving a cat is really helpful or if it will just make the cat unnecessarily uncomfortable. Ask your vet's opinion first. Lion cuts are pretty standard for Persian cats, like Jake's cat Thompson. Most groomers won't shave the cat entirely: the fur on the cat's feet, head, and tail all serve specific purposes.
If you decide to shave the cat, we recommend going to a professional groomer and not trying to do it yourself (anyone who has ever tried to give a cat a bath can guess why). There are cat specialists in New York, although most dog groomers also groom cats. Call around and ask. Paws in Chelsea was highly recommended to us. You can also check out Citysearch's list of groomers and this directory of groomers. Grooming the cat will likely cost between $50 and $80, depending where you go.