I have a friend who is paying out of pocket for in home care. How can she get it paid for and stay home?

Any resources that would help pay for SOME in-home care differs from state to state. I am not aware of any state that pays for in-home care 100% for assisted living needs. Any higher medical or full-time memory care needs would cost less at a facility.

A lot depends on the circumstances.
In most cases the only way to pay for care in your home is to pay for it yourself.
In some cases some help can be provided through different agencies.
Your local Agency on Aging might have some resources.
If the person being cared for is a Veteran the VA might be able to help a little or a LOT depending on where and when the person served.

gladimhere is right about Long Term Care Policy. Some do provide for in home care but the policies are expensive and once a person is in need of help, is Sick or elderly purchasing a LTC policy is impossible or cost prohibitive. My policy costs 5000 a year, I have had it for 11 years. Would I be better off if I had banked that and saved it for my care when it is needed? Only way to know that is to know when I would need it and for how long and since my Crystal Ball is broken I chose LTC insurance.

In home care is very expensive. There are no programs that will pay that, a facility is much cheaper. The only possibility I would think is it there is a long term care policy.

Looking at your profioe, your mom's care needs, with two amputated legs, is a very high level of care and you would likely not be able to find an agency to take this on.