If you go there just for hours building, you need to get an FAA PPL based on you JAA PPL (I think it's called a restricted FAA PPL because it depends on the validity of the JAA PPL) and an appropriate medical. I don't think for the validity of your JAA PPL it matters whether the hours flown and counting towards your experience requirements (12 hours PIC in the year before license expiration) were flown on an FAA restricted or your JAA PPL.
But if you want to expand your knowledge and get an additional license, particularly the UK-only IMC rating (which is not an ICAO license, after all), you need to go to one of the five-ish schools that do JAA training. AAA in California or OFT, OBE or Naples in Florida. Check the CAA website or one of the other threads for the definitive list.
If you're doing flight training (as opposed to hours building) you also need an M-1 visa and TSA clearance.
Florida weather is very reliable and predictable: there's a minor or major tropical storm about every five to ten days, april to october. A few of these grow out to be hurricanes and get their own name. But apart from these storms, an excellent place to go flying. And these storms are generally easy to predict. But if you're only going there for a week, you might just find that week coinciding with a tropical storm of course...
For myself, I would not want to go through the hassle of getting visas, TSA clearance and everything for just a weeks flying/hour building. Remember that just getting a visa appointment may take a month or so, and the appointment itself might take two-three hours plus the travel time. Spend those hours in an aircraft at your local field and you're halfway through your night rating, and a quarter of the way to your experience requirement.