It will probably be whoever typically handles the KLM & AF flights since that goes in and out of T5, which is where your international arrival would have to go. DL normally works out of T2 so I don't know if DL puts agents/crews over in T5.

Do you have onward travel scheduled on DL from JFK, that would probably help your case to get flown somewhere else. I had an experience in the spring where I arrived on an international arrival to JFK to find both flights going to my final destination on DL were cancelled due to weather/FAA control, leaving me with a 22 hour layover at JFK, no hotel/meals. I had no status at the time and convinced them to route me JFK-MSP, stay there with family for a day then go on to my original destination. The agents at the airport couldn't do it, because it was technically a stop over, but I talked to reservations and they did it without question. I just told them the flights I wanted. Won me over as a customer with that little IRROPS rebooking.