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Private caregiver search
This will probably be a bit difficult, but not impossible. You'll have to find a single lady without young children, needing a place to stay, with the right skill set that you describe. Hopefully, she will be strong and not too feeble or old, since caring for one person single-handedly can physically be impossible at the very end stages of dementia/Alzheimer's.
I have seen it work for a threesome that rented out one of my houses last year; the wife was a full time registered nurse, and went to work during the day, her husband did not work, and the gentleman being taken care of was 90 and in a wheelchair.
The nurse would then bathe the gentleman, with the help of her husband, and help him with medications when she was home. There were two of them to take care of him. In their arrangement, he paid for most of the rent from his savings. He had no family members that stepped in that I know of, and he passed away about a month ago.
I'm really not sure that only one person would be able to do it until your Mom passes away, since one person will need breaks, and will need to have vacations, time with friends and family, and alone time. I know someone that had two caregivers, one at a time, they would switch out every two weeks or so; there were still gaps in care/coverage since family emergencies and the caregiver's own appointments and affairs need to be taken into account.
When you say " in exchange for housing" , are you thinking that someone would stay with your Mom and take care of her for free? That's probably not going to happen, and sorry if I misinterpreted the type of arrangement you're seeking for your Mom. You also have to take into account benefits such as health care, dental coverage, etc., that people can get from working for a company, but can't get from working for you and helping your Mom.