Why not take the taxi/subway (1,2,3,A,C) to 34th Street-Penn Station for the LIRR to JFK via the Airtrain? The transfer at Jamaica should be very easy; if I recall correctly, there are elevators from the LIRR platforms to the Airtrain platforms. This way, you'll avoid the dreaded the morning rush hour traffic in Manhattan.

If you absolutely must take a taxi, I would follow the suggestion above and ask the taxi driver to take the Triboro Bridge to JFK and avoid the 59th Street Bridge.