Our favorite restaurant for our kids that age(s) is Becco. It is in the theater district and the food is a step above Carmines or Tonys. Your kids will love the pasta special that comes with an antipasto or salad first course. It will cost less than Carmines where the bill is always more than we expect.
The Met museum IMHO is a must. Don't try to do it all, just the Egyptian, Medieval armour (for the boys) and whatever else your kids would like.
After the Met go for a walk in the park (you are very near the resevoir)or even better...a bike ride.

I thought that Beccos had closed. (Never went since the idea of all you can eat is the antithesis of what I want - but know it can be good if you have bottomless kids to fill up.

Becco is open. You can order a la carte. Wine offerings are decent and moderately priced.

http://becco-nyc.com/

your kids might like Ellen's Stardust Diner in Times Square, the waitresses all sing broadway show tunes so its a fun time.

also in the Village you have to go to S'Mac, its very popular, all kinds of macaroni and cheese dishes.
Last time my husband took my daughter they ran into Chris Rock and his family and Chris Rock was very nice btw and spoke to them.