My background is in social work and I've worked with the elderly population. Unfortunately, I have never heard of any family members getting paid to care for their loved ones. There are home health workers out there but for 'everyday care' it is a private pay situation. I don't know your Mom's financial situation, but could she afford to pay someone to come in from time to time to give you a break?
What I've seen in my past work is that families will keep the person in their home until health issues and caregiving get to be too much. The family then makes the difficult decision about placing their loved one in an assisted living setting (depending on where it can costs thousands a month) or a nursing home. If your mother qualifies for Medicaid and is deemed to have medical needs serious enough for a nursing home, she can be admitted and have Medicaid pay. You might check your local county Human Services office.
Good Luck! I really feel for you. My 90 year old Grandma has Alzheimers and so I understand how taxing it can be on a family to take care of her.