Nervermind about the arrivals.
I just looked at BA's website and they don't have that in JFK.
I also forgot I have a RCC membership so I can go back through security (yes, I have a seven hour layover, ouch!) and chill there. If they will accept my Delta boarding pass at the security gate.
Or I can take up the offer to go in to the delta lounge in terminal 3 that baccarat_king offered. This is starting to look like the best option.
By the way, Thanks baccarat_king for offering. Much appreciated.