Because he was in international F on a connecting OneWorld carrier, the OP could have used the OneWorld premium lounge at MIA, nearby in terminal E. Not necessarily convenient to his departure gate, but an upgraded experience.

Admirals Clubs are considered "business" lounges, which in the US implies payment for food and premium booze. Separately, AA provides a small number of Flagship Lounges (Chicago, LAX, JFK, London) for international F customers. Admirals Clubs however are a pay-for-service, domestic business traveler lounge. International status and premium passengers are comped usage, but the vast majority of users of these lounges are business folk doing simple domestic travel, and they are accustomed to pay for what they consume (and bill to the end client). AA will provide free drink vouchers (as many as you ask for usually) to international premium passengers using the Admirals Club.

JFK is not designed for connecting passengers -- it can be done, but it is intended as an O & D airport. Changing terminals, as you experienced, is a pain. The best option in OP's situation is to fly MIA-LHR nonstop.