Transportation

According to the Bahn, your route would take you through Hamburg. From there you could go by train and bus to one of the coastal towns, from which you go by ferry to Amrum island. I couldn't find any car rental on the island of Anrum. I did find a car rental agency on an adjacent island, Föhr,, but you would still have to go by ferry to the mainland. You best options might be to either go by train from Florence to Hamburg, rent a car there, drive to the ferry, and take the ferry to Anrum. Alternatively, take the train and bus to the ferry, visit Anrum, then come back to Hamburg and rent a car there. This is a resort area, so one of the small towns along the mainland, where the ferry stops, might have car rentals. Or, just take the ferry, bus, and trains to Köln.

www.deutschebahn.de has the route. Since it takes 24 hours, Break it up by spending the night in Hamburg, then proceed to Amrum by regional train in the morning. You can get from there to Cologne by ferry/train. If you ned to rent a car, maybe do it along the way in say Osnabruck.
GermanWings has flights from Pisa to Cologne, but not everyday, and you'll still need to do an overnight somewhere (Hamburg?) to get to Amrum.