As mentioned above, there are no trains on the Amalfi coast.
You can take two routes from Positano:
1. Westward: via Meta di Sorrento > Napoli > Florence (basically west) or
2. Eastward: via Salerno > Napoli (with or without train change in Napoli) > Florence
Assuming you rely totally on public transportation, the earliest you can make it to Florence is 12:51 and whether you take route 1 or 2, the earliest you can make it is 12:51, because....the last leg to Florence, it will be on the same train, whether you catch it in Salerno, or in Naples.
The difference is...to make that train via Salerno, you need to depart from Positano at 6:00 am.
To catch that train in Naples you can depart from Positano at 7:10am, therefor you can sleep one hour more.
Salerno looks close on a map, but it's not. The road is tortuous and you need two buses. The first takes you from Positano to Amalfi where you change buses and take another bus from Amalfi to Salerno.
If you go the Sorrento way, you would take the bus to Sorrento (or to Meta) where you catch the Circumvesuviana commuter train to Naples, where you finally get on the high speed train to Florence (which is coming from Salerno).
There are also later options of course, that would allow you to sleep a little longer.
For example if you take the bus from Positano to Sorrento at 8:45, you can be in Florence at 14:51. But if you try to take the same train via Salerno, you would need to depart Positano toward Amalfi at 7:10.
As you can see, going the Sorrento way is faster than the Salerno way.
However, as Ken suggested, it would be a good use of your money if you took a private car with driver from Positano to Naples. Ken also suggested to take that car to Sorrento instead of all the way to Naples, but the savings aren't worth the trouble. If you hire a driver, you might as well ask him to take you all the way to the Naples station. The difference is between ~50 euro (to Sorrento) or ~90 euro (to Naples), but the second option saves you one hour.
In any case I'll give you all the resources you need to decide:
Bus schedule from Positano to Sorrento, or Positano to Amalfi and Salerno:
http://www.theamalficoast.net/orari_sita_bus_timetables.html
Circumvesuviana train schedule from Sorrento to Napoli (Naples)
http://www.eavsrl.it/web/sites/default/files/ftp_orari_ferroviario/Cy.pdf
Trenitalia train schedule for high speed trains to Florence from either Salerno or Naples:
http://www.trenitalia.com/
For the private car transfer from Positano to Naples there are tons of companies. Just google "private transfer positano" and you'll get lots of hits. In Italian you can google "noleggio con conducente positano" and you'll get even more hits. The price however is still around 90 euro no matter who you use. By car Positano to Naples (station) is about 1h20min if you leave a bit later in the morning after rush hour traffic in Naples is a bit less. Remember that as long as you make the 12 noon train departing from Naples, you get to Florence at 14:51, just in time for check in time at your Florence hotel.