Lucky you, Hulababy, New York is awesome!
I love the sound of the Hip Hop tours, Saladfingers, wish I'd known about them when I was there last year.
Things I'd suggest:
Katz's Deli is brilliant, but you should also try Yonah Schimmel's Knishery just round the corner. They serve knishes, these yummy potato dumplings (kind of like a massive, squashed potato scone), with all different flavours, and great American soda. It's a real experience!
Museum of the Moving Image in Queens. It's a big cinema and TV museum with loads of interactive exhibits (dub your own voice over Babe the pig, for example) and lots of nostalgia for adults.
New York Transit Museum in Brooklyn. Built in an old subway tunnel, you can sit in old subway cars from the 1900s onwards and it's great fun. Plus they have regular free talks/displays.
Economy Candy on Rivington Street, Lower East Side. I come here every time I go to NY. More American sweeties than you've ever dreamed of. Tin lunchboxes. Tshirts. And it's all dirt cheap.
As pirate says, The Lower East Side Tenement Museum is great too. They do fantastic tours.
John's Pizza on Bleecker St for the biggest, tastiest (inexpensive) pizza you've ever had, in a pizzeria which hasn't changed in about a million years.
If your DD likes reading, the Strand bookshop on Broadway is enormous and really good fun.
I love the Folk Art Museum at Lincoln Square - lots of strange old American art. It's tiny, but it's free and has a great shop.
Have a brilliant time!