Make her pick up the cat, take it to your car, hold the cat while you're driving there. Explain the cat will probably spend the rest of its life in a small cage with no one to love it and it is her fault. And have her hand over the cat to the shelter and take a tour of the shelter. I sounds incredibly harsh but pets especially cats live terrible and sad lives in the shelter. And because like you said she's not a kitten, she will probably never get adopted. Good luck. And you're not doing the wrong thing by putting the cat in the shelter so don't worry about it. The cat's fate is entirely in your sister's hands. And maybe you can show her how fun it is to play with the cat. Most people don't realize exercising a cat is quite fun.