Blimey you will be busy.
Fisrtly if you have a silver c have you looked at the possibilty of getting a course reduction. I dunno about Jar but under the old caa regs you could updgrade silver to PPL in about 25 hours instead of 45.
As for the rest you might do it if you are lucky. Florida isn't sunny all of the time and they do get some poor wx so you can't count onflying every day. Factor in the airrcraft going tech and a few booking messups and 5 weeks looks tight. May be better to get the PPL, do some consolidation and get the IMC when you return.
What did the school say about the timescales.