If you go to Florida (or California - there's a JAA school there too) it doesn't make sense to go there just to do a few hours. You're going to have to obtain TSA approval in any case, and arguably an M-1 visa too - depending on what you and the friendly INS officer at the border would consider to be your "primary reason" for visiting the USA. And you're going to need an FAA Class 3 medical to fly solo. That's a lot of work plus some significant cost for just a few hours of flying lessons.
If you decide to go to the USA for flight training, I think you should go all the way and aim to do your exam there too.