a walking tour would be a great way to see Florence in 1 day
One day isn't really that much so I wouldn't spend the day indoors. Florence is basically an open-air museum, much like Venice and Rome and lots of other cities in Italy.
I would enjoy the day walking around, taking them to see the sights. The center of Florence is really small, so from your hotel you can reach the main sights in less than 10 minutes walking. See the cathedral, the baptistery, and bell tower in Piazza del Duomo then head toward Piazza della Signoria and then onto Ponte Vecchio. You'll see many of the main monuments along the way.
I think for a 9 and 12 year old what would make the walking around very interesting would be hear about the historical details, intrigues and just interesting bits of each place that would make each location come alive for them. Show them the cathedral and as they observe it, try to have them comprehend that it took 200 years from when the first stone was laid to when the dome was finished - basically nobody that started worked on it ever saw it complete. If they've studied some art in school, they will see plenty of paintings in all of the churches and sculptures in the squares without needing to go into museums to see them.
There are a lot of artisan shops in Florence, artisans that have been making their products in the traditional ways that have been passed down through the generations. It would be interesting to visit them and see how they continue their craft. I think the best way to see Florence for your one day with your children would be to simple take a themed walking tour which ties each place together in one way or another, I recommend the ones offered by Vinarium here on their walking tours in Florence
page. If they have any special interests in art or science or history, there is one way or another that the visit can be tied together to make what they see interesting for them.
I imagine you'll be taking day trips during your stay in Umbria so if they find what they see in Florence interesting, you could always come back for another day