Do you mean boarding the Leonardo Express at the airport at about 13;30 and changing in Rome after that, or do you mean boarding a 13:30 train at Roma Termini or Roma Tiburtina after having made your way from the airport?
If the former, there isn't a direct train at that time from the airport to Florence, you will have to change at either Tiburtina or Termini.
If the latter, I respectfully suggest you have no hope. In order to get on a fast train at Termini at 13:30 you need to be in the station with at least 15 minutes to spare - say 13:15 - for walking time. The 12:53 won't work as you would only arrive at Termini with a fair wind following at 13:25, never enough time to get to the boarding area on the platform of the train you want. So the last Leonardo Express you could take would be the 12:38 which would give you 20 minutes at the Rome station.
I think it exceedingly unlikely that you would be off the plane, through the long walk to immigration, through border control (even if you had zero luggage) and then walking to the train station and onto the train in 38 minutes. That's if your plane is bang on time.
Sorry.