One option to completely avoid the van Wyck is to get a cab from LaGuardia to the Jamaica LIRR station via Grand Central Parkway to 150th street, then to Archer & Sutphin. Might cost you $20 or so, might take 10-15 minutes. You can then take AirTrain (runs every 5-10 minutes, takes 10-15 minutes, for $5) from Jamaica LIRR to JFK.
http://picasaweb.google.com/chris.j....AToJamaicaLIRR