How do I make a decision between a low-intensity skilled nursing facility and a high-intensity rehab center?

Isabel, the counting backwards and who is the President doesn't rule out dementia.

She needs a SLUMS (St. Louis University Mental Status) or MoCa (Montreal Cognitive Assessment).

Ask the hospital for a consultation with a geriatric psychiatrist.

Isabel, you've also posted about your mom grunting and pointing, and washing and re-washing clothes that don't need washing.

If mom is currently in the hospital, have you considered asking for an assessment of her current cognitive status while she's there?

She does live alone and is very sedentary. She did have PT before at home and would do exercises when they were there, but not on her own. They diagnosed her with AFIB too. She has had PT in the hospital. The first few days there, they could not get her to stand up unassisted. Now she takes a few steps down the hall with a walker. Still very wobbly.

Lives alone? Well, going home certainly not an option. The falls are likely due to balance issues. My brother did well in rehab with balance exercises for certain, and while it was not "high intensity" it did help him; problem was he didn't continue to balance exercises once released. There would likely be some strength building exercises for your Mom, getting her up, deciding what equipment will work. I doubt a skilled nursing facility would help Mom at ALL.
It sounds to me like it would be the usual 21 days in rehab, hope for some improvement in balance and strength and some training in actually getting movement going, deciding what assistive devices.
Throw yourself on the mercy of the discharge planning team and social workers at the rehab right away, and it is looking like ALF next step if your Mom can get on her feet a bit. I doubt she will be able to return home from either facility unless things change drastically in that time.
I wish you the best of luck. Work with the team there, attend some training sessions in PT and OT if you are able.
Our balance gets just dreadful with age. I saw my bro go through it and I see it in my 80 year old self. While I weigh more than I ever did at 150, on a bus I feel like a feather in the wind flying about if I am not holding on, which is purely a matter of balance. Have had two falls in the last couple of years and while they were do to my carelessness with garden implements and walking backward while cutting my brugmansia, etc. they were ALSO primarily a matter of being unable to "catch" myself--the balance. I am greatly helped by balance exercise, when I remember to do them!!!!! (hee hee).

Acute rehab is generally pt/ot twice a day 6 days a week.

Can mom handle morning and afternoon sessions?

Can mom remember to do those exercises independently?

Is mom motivated to improve?

Can she handle the "high intensity" rehab?
With the "mild dementia" will she be able to participate in rehab AND will she or you be able to continue the therapy once she returns to her home? (does she live alone? What is her activity level like?)
Did she have any PT or OT after the first, second, and subsequent falls?
I think with the 5th fall in 4 - 5 months she may continue on this downward spiral.
What else is going on? It seems odd if it was a "simple" fall that she would be in the hospital for a week.
Have they done any PT or OT while she has been in the hospital? How has she done? And if she has had PT have you observed how she does?