Abandoned cat? HELP PLEASE!?

My mother and I had to kick out my father and my half sister a couple months ago. Here's my question. She left her cat here. Now she was a horrible cat owner. She never cared for the cat and was very neglectful. I love the cat and want to keep it desperately. She claims "the cat was her mothers and now it's all she has left of her mother!" That's not true at all but I digress. We have been caring for and feeding the cat for over three months now and she is trying to take it from us... what can I do? Do I have a right to keep the cat? Before she moved in with us she used to do drugs with the cat in her apartment and the vet said this might have led to the cat's possible neurological problem. I really want to keep her I love her to death... WHAT DO I DO!!!!???? HELP!