Texas native here (San Antonio) who now lives in the desert outside of Tucson. I'm also 76 with bad knees.
I agree with those who say to get lodgings with air conditioning. In the summer, we keep our thermostat at 77° or higher and use our ceiling fans, so we are not in the truly refrigerated air crowd. That makes typical European AC just about right for us. I'm not fond of open windows due to the lack of screens and the noise.
Even if the price is higher, a place with an elevator will be well worth the money. Has your mother ever gotten any shots in her knee? I get shots in my knees before each trip, but my osteoarthritis still requires that I do stairs one at a time, making me quite slow. Keep that in mind because as hard as you try to avoid them, there will still be places where she will need to go up or down a few steps.
You've been told about the accessibility options for the grounds at Versailles, but research it online to see what might be possible inside the palace. Even with a timed ticket, you will still need to wait outside to go through security. There may be some other options if she's in a wheelchair.
Where we waited was in a long line on gigantic, very round and slippery (it was raining) cobblestones. I was not wearing the best shoes for that and I thought I was going to fall as I minced my way across them.
That experience ended my days of trying to be fashionable anywhere and now I wear lace-up, waterproof, breathable ankle boots exclusively on my trips. They have kept me from turning my ankle and falling on cobblestones or broken pavement on more than one occasion.
My favorites are Keens like these. And these exact Danskos.
I wear them with Sockwell Moderate (15-20mmHg) Graduated Compression socks which keep my feet cool or warm as needed and dry, as well as help fight fatigue. They're not cheap, but there are many styles to choose from and you should be able to get to them with this link.
I agree with those who advocate taking the bus instead of the Metro whenever possible. I love taking the bus because I'm never in a big hurry and I like seeing where I'm going. Remember that you will need to take a local train to Versailles and unless you take a taxi for the short distance to the waiting area, it is a bit of a walk that could be challenging for your mom.
At Versailles there is a very nice restaurant on the grounds. You must make reservations for it. That's one reason to get a golf cart or take the shuttle around the grounds.
Another is to get an idea of the size of the place. It's immense and riding around certainly made me understand why there are drawings and paintings with people on horses or in carriages, as well as how there could be hunting there or how easy it would be to sneak off for a tryst.