I've been to Paris ten times, but not lately. My last favorite hotel was Louvre St. Anne, which had a triple on the top floor, an elevator, and new bathrooms and A/C. Have been there twice. But there's no chance you'll get in to such a small place for next month. Not far away, I wonder if the elderly but renovated L'ile de France and associated Gallion Opera (near Opera Garnier, much better location than Rue Cler, have air conditioning. They are THREE STAR (very basic) tourist class hotels, old style France travel. I'd ask for a courtyard room so it's quieter. PLEASE CHECK ONLINE REVIEWS! But they might be in your price range, and no worse that the Leveque. Haven't been to them in 35 years though. Our Hilton in Paris has new management and branding, and would be over your price range.
I personally hate Rue Cler, and despised our one-night stay at the Leveque, could not sleep because of heat and noise. But I know many other Rick fans love it. Use the Search box top left to find other recent threads with Paris and Hotel in the subject line.
Do you understand that street crime (other than possibly pickpocketing) is far lower in Europe than it is in most US cities? You do not have to worry a lot about safety in downtown Paris. In the far suburbs and distant, lonely areas, maybe. When I first went to Paris, around 1987, I was entering the Metro when an army commando team with long guns jumped over the turnstiles for their routine anti-terror sweep of that station. This was before 9/11, although after Lockerbie, I think. Disclosure: I was born in Manhattan and grew up there.
Edit: I wonder if your mother needs a modernized bathroom? There is a certain Paris tradition of tubs elevated on legs that could be difficult to step into, with no grab bars in some cases?
https://community.ricksteves.com/travel-forum/france-reviews/paris-hotel-recommendations-for-senior-citizens