The European sale is your best option, it ends on Wednesday 12 June. Don’t feel too hard done by about missing out on the exUK sale, as rtn fares to MIA were £2,400 in Club and £1504 to JFK.
On BA, your options are to fly from from Heathrow to Newark, New Jersey or from London to JFK using the following routings:
London CIty- JFK (CWLCY A318)
In addition, there is the option of Open Skies which has an excellent Premium Economy offering:
Paris Orly-JFK http://www.flyopenskies.com
To Florida, there are several services per day from Heathrow to Miami, and from Gatwick to both Tampa and Orlando.
For your connecting domestic flight to NYC, do consider smaller airports like Fort Lauderdale which can be a lot easier to negotiate than MIA.
Those other options aside, exEU fares are probably your best bet, with return prices around €2000 rtn (About £1700):
You should be able to get CWLCY for under £1500, the best rates I have seen are from Sweden.
Use a reward saver to get to your starting point, once you’re there you may as well do as Travelwave suggests and enjoy a day sightseeing in your chosen exEUR destination.
If you haven’t tried CWLCY, I’d recommend it.