Sorry to sound negative but I think your chances of making it are slim if you have bags to check.
At LGA lately the baggage wait can range from 10mins to 55mins. Moreover, you will be travelling in peak rush hour (1hr travel time seems about right) and most airlines have a 1hr cutoff for baggage acceptance.
Best care scenario:
Arrive: 0735
At Baggage claim 0745
At taxi/bus stand 0800
At JFK check-in 0905
Cheers,