We were there last summer with our 12yo boy, also not into sports, very bright but museums, ruins, etc were not his cup of tea. We did one museum every morning (we are morning people) and then had a list of possibilities for the afternoons, and then in the evenings we would walk around the river area. We were there for less time, so had less unscheduled time to consider, but we considered going by train to Lucca and renting bikes and riding around the wall that goes around the city. Also considered training to Pisa, just for the fun of it being something the is familiar with. There is a public pool (there are actually 3 - if you google visitflorence.com, they have a list of family ideas and include descriptions of the three pools. The one we were considering is next to the river just south(?) of the train station [I think, I don't have a map in front of me]). We didn't go to the pool, but it was in our pocket if we needed it. We also strolled around Boboli gardens one afternoon. The Mercado for lunch was also fun - we walked there from/after the Academia visit...
He enjoyed the Academia, really didn't like the Uffizi, enjoyed the Dome climb and liked the rest of the features associated (although he did not come with us to the Duomo museum). We were on a 5 week trip with his two sisters, and sometimes we would leave him at the rental (apt. in our case) with a Netflix movie on the iPad. I know, the horror, but we battled for weeks making him go to all the places because I felt so strongly that it was important for him in the long run, but after a few weeks, we realized there was a balance to be struck that made life more pleasant for everyone. His older sister loved the things we did, he did not. So worst case scenario, there may be an afternoon when he could use some down time and do this.
