Definitely stay in NYC and take advantage of all the city has to offer.
Museum of TV and Radio is neat: http://www.mtr.org
and a must see is the Museum of Natural History: http://www.amnh.org
I happen to like the Museum of the City of New York, http://www.mcny.org
Of course if they are into art, there's plenty of places, such as the Museum of Modern Art http://www.moma.org, The Metropolitan Museum of Art, http://www.metmuseum.org, the Guggenheim, http://www.guggenheim.org and plenty more.
Take them to Soho (means "South of Houston") and the East Village too. Central Park is always worth the trip http://www.centralparknyc.org. I would also have them see Grand Central Station - its unlike any other train station in the world: http://www.grandcentralterminal.com. They have very fantastic tours of the station.
For 9 and 13 year olds, they would probably like to see the Apple store (there's one on 59th and 5th Ave and one on Prince St in Soho). They can pick up the latest iPod stuff for about half of what you would pay in the UK. And while the kids are at the Apple store, the wife can shop at great stores nearby. With the dollar in the crapper, you'll get great deals here.
Let me know if this helps.
Once again I thank you all for your help...
It'll be great to see the new arena. The first fan to spot us gets free beer! (Not saying which games we're going to!)