If it's just his tail and fins, I would say he's going to be alright with time. I can only guess that the water was much too hot and that he may have damage from the chlorine. Did he hit anything in the sink, like the drain, that may have caused damage? You can use clean, warm water, along with an aquarium salt treatment (10 days only) to help heal and stave off infection.
My female jumped out of her tank and spent at least a half an hour on the cold hard floor, her fins and tail were a mess, but with time, and clean, warm water, they grew back. Looking at her now, you'd never even have known that she jumped out.