It is possible - but it will take some planning.
For one - between Miami and Key Largo lies Biscayne National Park. PWCs are banned from the park. You can not go south in Biscayne Bay past Chicken Key. If you do not know where that is - look for the two blue smoke stacks that are made to look like lighthouses south of Matheson Hammock. This is Old Cutler power plant - which is right next to Chicken Key. That is as far south as you can go in the bay.
So you would need to either trailer to Key Largo OR head offshore. The eastern boundary of the park is the reef line, from yellow marker "N" to the north to whistle buoy red "4" to the south - including Fowey Rocks, to just past Pacific Reef. You have to be outside this line - in theory. In reality you may be able to run Hawk Channel - but the parks service could harass you if they wanted to. Also - watch for the numerous patch reefs that are very shallow at low tide. Some patch reefs are partially exposed at low tide - you can run aground.
Here is a link to a map of the park. Note that the map is not intended for navigation purposes.
http://www.lib.utexas.edu/maps/natio...ks/bisc_95.jpg
Once past Biscayne National Park the ride will be easier. You can run along either the ocean side or the bay side of the Keys. I am not aware of any major restrictions there. But you will still need to study charts of the area. You will also have to know about speed restrictions and whatever else there is.
As for places to stay - the keys has a lot of hotels with boat access. There are also a ton of water front restaurants and just as many marinas to fuel up at.
The trip is very possible, and would be a fun run. Plan a couple of days for it. And remember to plan for the trip back to Miami - unless someone is going to meet you in Key West with a trailer.
The distance you can plan to run in a day depends on conditions and your endurance. I would make a few runs from Key Biscayne to Chicken Key and back to Key Biscayne (be careful to keep out of the park!!!) to see how much I can run. This portion of the bay is a good place to run because it can get choppy, and would be a good simulation of a Keys run. The bay side of the keys tends to be calmer than the ocean side, but it can still get choppy in exposed areas.