NY La Guardia to NY JFK isn't that far by road. I know because we travelled from Manhattan to JFK and went straight past La Guardia (the downside was that we got to JFK to find that we should actually be at Newark, but that's a different story altogether!!!). From recollection, La Guardia to JFK is around 15-20 minutes depending on traffic, but double check this.

I think I'd go along with AndrewP and fly direct into Miami from Heathrow with AA and then drive up. The drive isn't too bad and there's lots of service stations to stop off at along the turnpike.