Since my mother‘s name is on the house, as well as mine, what would happen to profit from a sale?

I honestly cannot imagine why your mother's name was put on this home.
I would see an elder law attorney about this. It no longer qualifies as do-it-yourself guess work, because doing that has landed you here now.
Get help that is professional in your own state, and own county. I wish you the very best of luck.

In a message you posted more info:

"She IS eligible for assisted-living with memory care. I have it all arranged and she should be moving shortly into an appropriate facility where she will be close to me and I can participate in her health hopefully improving it or at least giving her as much love as I can before she forgets me"

Do you have FPoA authority for her? (read the document to see if you can legally sell her half of the home). I'm not sure if your disability status makes for a different situation. The answer to your question may be state-specific, plus it's regarding an in-home Waiver program. I think you should ask this question to a Medicaid Planner for your home state.

We got the house in 2018. She’s been in a SNF for almost 60 days. Makes her eligible to be on the assisted-living waiver program. The details are a little difficult, but I’m learning as I go along.

Do you mean that mom was a co signer in case you could not pay off a mortgage?

I too don't understand why you had to put Mom on the deed. I can see having her co-sign for a Mortgage, but that does not make her an owner unless the mortgage company or she wanted her on the deed too. Is Mom helping you pay the mortgage?

You cannot get Mom into an assisted living or memory care if she has no money. Medicaid will only pay for Long-term care. Unless, ur State has vouchers and then I think they have limits, only so many are given out.

If Mom goes into a facilty, the home is exempt. As co-owner and a disabled child, you can remain in the home. Upon Moms passing, the home is no longer exempt and a lien can be placed on her half of the home. The lien will not need to be satisfied until u pass, leave or sell the home.

Most states' Medicaid program only pays for LTC, and not AL or MC. What state are yoiu in? Check online to see what your state pays for.

Medicare doesn't pay for custodial care.

https://www.medicaidplanningassistance.org/medicaid-estate-recovery-program/

Sorry, I have a lot of questions:

Could you explain "I had to put her name on my house because I was 53, I don’t get that because I am disabled."? Did that just happen? Was it part of Medicaid's 5 year lookback? If not,

Every state has utility and other assistance programs (including repairs to bring it up to code); have you checked those out?

If you sell, where would you go? As long as you're in your home, the SNF can't touch it. But if Mom's name is on the deed... I believe the rules vary by State.

Medicare will only cover 100 days in a SNF after a 3-day hospital stay so Mom's care would be covered entirely by Medicaid.

You could do with a benefits counselor at your local Area on Aging office. They're there to help families know their options for care, housing, etc.