I am not a parent, so read accordingly! Firstly, this is certainly a tough one. Depending upon the resources you have (financial and physical) and the children's personalities ...
The idea to see something of what is going on sounds a positive starting point. Perhaps a mini tour of the terminal you are in? This will depend on how much hand luggage you have and whether the children are likely to want to explore.
Most terminals have lots to look at (possibly TOO much!) and I have never been into the Delta Terminal at JFK. You can get an inter-terminal bus that runs a 'loop' around all the terminals and that gives you a tour and yet brings you back to where you started. You might find some details here:
http://www.jfkairportguide.com/
Secondly, consider using a paid-for lounge when they get tired. These can be expensive but have proper seating and facilities. However, your children would need to know that they have to be quiet and it is not a play pen. But, if they have been running around the terminal and maybe taken a bus ride, they might be ready for a snooze.
I do know that, most times you plan something for children, they manage to make it something else!!