How long do you plan on staying in New York ? Are you flying in the day before, leaving right after or plan on being in NYC for like 5 days ?

There is also a Staybridge Suites close to Times Square with kitchens/frig if the apartment falls through.
http://www.ichotelsgroup.com/staybri...ee/hoteldetail

If your planning on staying more than a couple of days maybe fly in the grandma the day before and shorten her stay while getting your daughter a taste of NYC. Otherwise, I might drop her off in STL if we're talking 2 or 3 days.

Cheers.