Compensation for care from a friend?

That's the number one question asked on this site. Most family and friend caregivers provide unpaid caregiver support unless the patient can pay them from their funds.
Here is an article that explains some of the alternatives that you might qualify for - Medicare does NOT pay for 24/7 caregiving in the home in most cases. Medicaid will pay for some caregiving by waiver in the home but the lists are long; it's not 24/7 other than a very few places like New York City; and it's really not a living wage with benefits. Any money to help pay your friend will be considered taxable income and may affect any household benefits they get if low income.
https://www.agingcare.com/articles/how-to-get-paid-for-being-a-caregiver-135476.htm

You can form a contract with anyone you like. It might be a good idea to download pro forma / template contracts from agency providers so that you can see what sort of things need to be thought about and decided between the two of you.

I'd also just check with your local health and social care regulators about things like insurance, minimum wage rates, tax implications and so on. The trouble all starts when a person is paid for their work, that's when government agencies tend to feel entitled to take an interest in what's going on. It may be that they won't interfere with a private arrangement in a private home, but still it's best to check.