Here's an article about short-term stay apartments in NYC. Maybe some info in it would help direct your search.
http://www.nytimes.com/2012/06/03/re...apartment.html
Have you looked on airbnb.com?
I imagine that your budget is the big issue here, since short-term stay apartments are much more costly than regular apartments, and $3000-$4000 is not that much in Manhattan. Riverdale should be much cheaper than Manhattan, but don't know if you'd be able to find a short term deal there.
Good luck to you and best wishes for your daughter's health.