Yes, I know we don't want to abandon our parents, but the cost of living is expensive enough. Adding the additional expense of providing for your elderly parents means you are not able to save for your retirement or cover your own expenses. The end result means that one day, you could be a financial burden on your kids.
In this situation, I would advise you to look into a short sale. Since this is your mother and late father's second home, she may have another home with equity, or she may have other assets. Those assets could be in jeopardy if she lets the house go into foreclosure, and she might not be able to protect all the assets if she files bankruptcy.
Foreclosure is a dangerous option because, depending on which state she lives in, she could be on the hook for any unpaid balance after the foreclosure sale. For example, she might owe $200,000 on the mortgage, but the house sells in foreclosure for $150,000. She could be liable for the $50,000 difference.
Bankruptcy may also not be an option since it seems she has other assets. You state this is her second home, so her primary residence could have equity. She definitely needs to work with an experienced bankruptcy attorney before exploring this option.
A short sale removes your mother's name from the property title and the loan, which would also resolve any future issues with the bank.
You will need to have a valid power of attorney, or POA, to handle this process for your mother. If you don't already have a POA in place and she has late-stage dementia, doesn't recognize you and can't handle any of her affairs, then you aren't going to be able to work with the bank. For obvious reasons, your mother cannot execute a POA when she is no longer lucid.
If she's in early-stage dementia, forgetful and unable to manage her personal affairs, a local estate-planning attorney would be able to help you. That attorney would need to meet alone with your mother and assess whether she is aware that she is giving you this power to communicate directly with the bank.
I hope you were able to address this early enough and can resolve this easily with a smooth short sale. I assume your mother never wanted to burden you with her troubles. Good luck.