Well, you have a few options, none of them great.

The first thing I'd try is calling one of the local car services... I bet it's not too outrageous (~$50 maybe??? But really not sure.)
Two companies that have been around forvere and are reliable are at:
1-212-777-7777
and
1-212-666-6666
They generally just want you to call when you arrive and they're waiting for you when you get your bags, but give them a call to check their rates

Other options are:
Airport Bus, info here. About an hour, $13.

Local bus, subway, bus combination - Q48/LGA terminals to the 74th/Roosevelt subway station to the Kew Gardens subway station to the Q10 bus to JFK terminals. $4-6, 2-3 (4?) hours.

OR

Local bus, subway, AirTrain - as above, Q48 to subway, but go to Jamaica Subway station and transfer for the AirTrain. 1-2 hours / $9.

Hope that helps.