Are there any government funded programs for my mother so she can stay at home and take care of our grandmother?

Medicaid often helps some. However, each state is different (these programs are federally funded but state administered). So, try going to your state Web site. Type "aging" in the search and follow the local links for information. You will find the National Caregiver Support Program on the site (could have a different name). This can also be very useful. Good luck.
Carol