Homelessness is a rising problem among seniors, a vulnerable population. Do everything you can about not going there.
No, it's not your problem, but you can be a decent human being about it. For all of those who say the obvious "we're a business, not a charity", this could be YOUR mother or grandmother. All these "sweet granny baking cookies and babysitting" and playing bingo at "the old folks center" comments are stereotypical nonsense. Many of today's 80-year-olds may be on Social Security, but they may not want to live with their children and grandchildren and do their babysitting because they may have their own lives, don't get along, or don't have the patience. Their families may not want to live with them, either. Additionally, they may have difficult health issues or their younger family members may have health issues or can't afford to take anyone else on. This is 2022. "Old" is relative. She very well may have something better to do with her life than play bingo.
My adopted aunt chose to go to work on her 80th birthday. (She worked part-time at GNC.) The trouble started when she was 83. She was living with her disabled son and a grandson who was on the spectrum. He lost his job, and they couldn't afford rent on the mobile home they were in. Social agencies are stretched horribly. They ended up living in a shelter for months (she slept on a lounge chair in a church basement!), and while everybody got them a motel room once a week, her health went downhill, even though they've been in a Section 8 apartment for 18 months. The stress took a lot out of her. It's been a struggle to see what has happened to a vibrant, working older person.
The first thing I would do would be to give her the respect and talk to her honestly. I'd suggest you explain that you simply cannot afford to rent the apartment for that amount of money because you're going to need to pay your mortgage and other bills. Ask her how you can help her resolve her housing situation. Maybe you can compromise between your number and her rent (we've made compromises for seniors on a fixed income, as well as military), but maybe not. Depending on her health, ask her about her preferences. Don't expect a family member or a social service agency to rush forward to save her. If she belongs to a place of worship, they might be able to help. I also would suggest that you contact other landlords in your area. Maybe they know of something. Also, ask around to see if there are any older realtors/investors with ideas.
Roommates can be tricky. They may not get along, and that's more stress, which is particularly hard when folks get older.
I wish you luck. It's a difficult situation to be sure. I don't know much about the Ohio market, but could there be any mobile homes that would work for her?
Bless you for trying to help. You have a conscience.