IANAL. Some of this pertains to my experiences in Texas, which may be different from your situation. But similar provisions probably exist elsewhere. Also, you're a bit vague about the facility.
On the assumption you mean some kind of nursing home or clinic:
If your mom is that bad off, what you really need is to get her identified as disabled, if you have not already. Then you work with the staff at the facility to get her moved in and have her apply for Medicard/Medicare, the works. Because, unless your situation is special, it is not you who needs financial assistance but your mother.
What usually happens in this situation is that the facility eats a small part of the first month's bills, then they get paid out out of SSI/SSDI/Medicaid/Medicare. The typical arrangement is that they become the payee for the person's Social Security, and they get any other financial resources the person would have acces to, in return for letting them stay as a resident perpetually.
On the assumption you mean some kind of long-term facility that is primarily medical care and not a living arrangement:
Again, trying to get Social Security is the first step and you should at least try to talk with a lawyer, even if it's just a brief, pro bono consultation. The facility should have a social worker who may be able to talk to you about financial assistance. Now, this is where losing the house may have entered the picture, as the facility would probably like to be paid up front and on time. But barring special circumstances, you don't have a legal obligation to financially support your mother. If your mother requires living in the facility for medical reasons, she is probably disabled, and if she is disabled, you should be able to work things out where the facility gets paid.
If she owns the house and a transfer of property is a problem:
If the house is being counted against her expenses for a number of months, as is sometimes the case, and the house is technically hers, one option may be to go ahead and transfer the house to one of you, then go into debt for however long her expenses will not be paid. It is likely going to be painful, but a) it might be cheaper than losing the house depending on circumstances, and b) ultimately it's a temporary problem, albeit not one of short duration. Since the house is on the line anyway, you might as well see if you can use it as collateral for the needed money.
If the problem is that your bother is using her Social Security to keep a roof up:
Then you guys probably have some hard decisions ahead.