You should be able to get into the lounge anytime the day of flight, so even nine hours before should be OK. The question is how far in advance they'll let you check-in your luggage at JFK in order for you to be able to go through security to get to the lounge. I vaguely remember once having an issue with trying to check bags more than four hours before departure, but don't remember for sure. Maybe call or tweet DL and ask.
In any case, worse comes to worse, you can always hang out in the LGA Sky Club after landing for a while, then head over to JFK once it's closer to the flight to TLV and then go to the Sky Club there. Personally, I would recommend stopping at one of the restaurants in Kew Gardens Hills on the way from LGA to JFK to eat something. They're not more than five minutes out of the way.