I very much understand the struggle to cope with a difficult parent and the need to individuate. It is time to have more of your own life and less intrusion from your mom. This will not be easy, but it will be important to do.
Your mom will not understand — so the work here is to have compassion as you separate. In 12-step programs they call it detaching with love, but I think the real leverage is to find the compassion for your mom as you unhook from her. This is not something you are looking for her permission to do — and you most likely will feel guilty for having less contact. But this guilt will be easy to manage compared to the resentment you are feeling.
Limit your contact while you help to find social services for her in her area. She needs people around her on a regular basis for support. You can help find those services and help coordinate getting her there initially as you reduce your direct connection. Take her staying with you off the table altogether. If you don’t make the boundary it won’t happen. In the meantime you may want to have some therapy for you and your sister as you initiate this plan.
It is always better to choose feeling guilty than feeling resentment.