How do I apply for financial assistance for taking care of my elderly mother?

Generally, whether or not free or low-cost services are available depends on your local area. Some areas have services supported by the state/county/local charities but many don't. I know in my mother's area, there is an agency that provides those types of services but you need to be eligible for Medicaid to receive them, so my mother is not eligible.

My mother needs the same types of help that yours does. I've been doing it (for free) with the help of my sister for almost 7 years. The last two years, my mother has had a housemate who provides some assistance in exchange for free rent. My sister and I still accompany my mother to all her medical appointments and take her shopping at least once per week.

If you're looking for an agency that would pay you to provide those services, that's not available in most places. Most people who receive pay for taking care of family members are paid directly by the recipient. Sorry to be the bearer of bad news. Most people are surprised to learn how little help is out there, and it is quite appalling that families are expected to manage all this without help.