You can search for 2 bedroom hotels in NYC. There are a few. I
n addition to what's been mentioned, a couple of the Hyatt Places in NYC have 2 queens or 2 doubles and a pullout. Look at the Leo House. It's basic, but it's comfortable and they have one or two family suites. I'd also look at the Beacon Hotel and the Newton.
With children, I wouldn't stay outside of the city or at the airports. The transportation time would become a burden as would the cost. Long Island City, Brooklyn, yes plus, maybe, Flushing. Don't do New Hyde Park
As said above, timing is everything. When are you looking to come?