Where to spend an afternoon in 110F Florence?

Visit the Duomo and San Croce church.

Doug, which other museum do you plan to visit? If not on your list, the Bargello is highly recommended. San Marco is very good as well but ticketing closes at 1:20 (13:20). Maybe tomorrow, if you're still in Florence? Very close to S. Marco is Basilica Santissima Annunziata (free) and the historic piazza of the same name: recommended for at least a walk-through.

https://www.visitflorence.com/florence-museums/bargello.html

https://www.visitflorence.com/florence-churches/santissima-annunziata.html
https://www.visitflorence.com/florence-monuments/piazza-ss-annunziata.html

https://www.visitflorence.com/florence-museums/san-marco.html

I would personally skip the duomo itself as I wouldn't want to deal with standing in the queue to get in if it's 110F.

We visited the Duomo yesterday. We’ll go to the Academmia tonight or tomorrow morning, and today we’ll also go to the Galileo Museum. I’d like to check out the Bargello Museum but I have to be mindful of my two kids getting over museumed. I’m thinking they’d love an hour or two of cooling in a pool or something.

Here's a post on public pools in/near Florence. Give yourself some time off and chill (or as best you can in 115F)https://www.visitflorence.com/what-to-do-in-florence/pools-in-florence.html
Article gives directions by public transit. Just be sure you double check the hours, etc before trekking out. Also check out swimwear criteria and if hair cap is required. Sometimes there are differences to US pools.
I also tend to like churches, crypts and basement things for their coolness, but not sure whether it would creep out the kids. The Palazzo Vecchio has kids activities and has basement tours that should be cooler than some building with no AC. Here is the link of kids suggestions http://musefirenze.it/en/musei/museo-di-palazzo-vecchio/

Also check out swimwear criteria and if hair cap is required.
Sometimes there are differences to US pools.

A really good point from Maria: I just checked into some basic info for a pool in Florence and swimcaps are required for all. You'd also have to buy/rent a padlock for the locker. Doesn't look from pix like men/boys have to wear tight-fitting type suits, as can be required for some pools in some countries.

http://www.lepavoniere.it/piscina/
https://www.tripadvisor.com/Restaurant_Review-g187895-d2269111-Reviews-Le_Pavoniere-Florence_Tuscany.html

We didn’t get to the Galileo or Bragello museums before they closed (1:30 and 2:00 respectively).

We might just chill out in our apartment, go for slow stroll a bit later, and then go to a movie this evening. I’ll see if the kids are up for a swim, thanks for the info.

We’ve been totally ok with the heat but it is nice to get out of the mid day sun.

The Ferragamo Shoe Museum is really interesting....and air-conditioned!
I went there myself to get out of the terrible heat a couple of years ago.

Have you considered going underground? I found these:
https://www.viator.com/Florence-tours/Underground-Tours/d519-g12-c32760