Can the nursing home force me to sell mother's car?

My mother has dementia, due to several falls she is now in a nursing home and we are starting the paperwork for her to stay there, she is 86. There is only me and my sister we are both in our late 50's. There is NO power of attorney, no will, nothing... she has always been in charge of her own affairs. She has a 2010 ford with low miles around 11,000. the title is in her name alone as she bought the car and paid for it. let me add the car has been in 3 accidents each over 5000.00 in repairs and right now it has some dings and dents.The nursing home is telling me I have to sell it for kelly blue book value. I do not sell cars I told them the title was in her name and i can not sign the title. The lady is telling me they will get a notary to come in and sign it over so i can sell it. I do not want the car i will be glad to give them the keys and the title i want no parts of it but they are insisting. My Question is can they force me to sell the car?