Actually, a green ig's tail SHOULD grow back, albeit it will never come back quite as long or nicely shaped as it originally was. It's much easier for young igs to fully regenerate their tails than older lizards. That the tip is turning black and she has a depressed appetite highly concerns me. As with any herp, you need to double check that her habitat is correctly set up and that humidity and temps are correct. I would definitely advise taking her into the vet and have them examine the break point if you haven't already. If the tail become gangrenous the vet will need to amputate more tissue to prevent the infection from spreading and killing your lizard.
Oh and since you asked before, I found a couple videos regarding green iguana taming. The trick with igs is persistence and patience.