Hands down the best location for an apartment is the Radio City Apartments....right in the heart of midtown close to the Rockefeller Center (Top of the Rock is far better than the Empire State Building), Fifth Avenue, MoMa and walkable to Times Square, Central Park, Bryant Park etc.
They have apartments in various configurations...I haven't personally stayed there as I live 35-45 mins from Manhatten...however I knew a Swedish cabin crew member for SAS airlines and they all like to stay in the apartments there when they get a layover in NYC for a few days and highly recommended it to me for when groups of friends are over from the UK.
www.radiocityapartments.com
(the larger family apartments understandably get booked out very quickly, so if something comes up for the dates you want, don't dither and book immediately!)
You can book direct with the link above, and some websites will let you either book now and pay later or have a no penalty cancellation fee....check first though before booking! A good place to look for price comparisons is www.trivago.com (not sure if there is a Trivago.co.uk but I know that I've seen it advertised on English TV stations).
There are people coming from all the the world as well as domestic US visitors and in school holidays individuals and families are coming over from everywhere.
Whatever you do, ensure that you have a good travel insurance policy, not just for medical but for cancellations, lost or stolen items etc. and best to buy your travel insurance within a day or two of booking your trip!