I don't have a direct answer to your question--we checked into a trade for NYC and it sounded like it was going to be a long shot, so we made alternative plans. Hopefully someone will have good news for you.
If you aren't able to do a trade, for a family of 5, we'd definitely recommend the Doubletree Guest Suites which is right on Times Square. We just returned from a trip to NY right before Christmas (and thankfully before any snow), and it was wonderful. The Doubletree is highly rated on TripAdvisor, and we found them to be very friendly.....especially helpful when we pulled up and weren't sure where to stop to unload and the doorman ran out and stopped traffic on Times Square during rush hour so that we could back up to the right area! We had a two room suite--2 double beds in one room and Queen sleeper in the living area; separate empty fridge in addition to the mini-bar, microwave, large bathroom with 2 sinks. It was pricey but huge by NY standards, and the location right on Times Square was VERY convenient for both the Theater District and sightseeing. Anyway, just wanted to mention it in case you don't get your trade. I know hotels are a challenge with a famly of 5!
Good luck!