Are there any discount or points programs you could qualify for to bring a nicer hotel cost down to what you want to spend?
North Jersey is fine to stay in, but I'm not sure how much you'll save if you're public-transiting to the City every day.
Anyway, I usually day-trip it with DD and if we stay overnight, we stay on the UES with my brother, who has floor space.
We like to go to the Met (new admission policy is $25 admission for out-of-state adults), the American Museum of Natural History (still a "suggested donation" to get in if you don't care about seeing IMAX or planetarium shows), and the New York Historical Society (which has a pay-what-you-wish late night weekly on Fridays.)
The Marriott Midtown East is fine; you'll need one cross-town subway or bus to get to Times Square, but a lot of what you might want to do is walking distance - St Patrick's Cathedral, Rockefeller Center/Top of the Rock, NYPL Schwartzman Building (the one with the lions), Grand Central Terminal, Empire State Building, Chrysler Building, AG store (not sure how old your DD is), etc. The Met is a short uptown subway ride away.
What might you be interested in seeing with your DD? There are a lot of great musicals currently running, with a revival of Carousel and My Fair Lady set to open sometime later this year. And don't ignore off-Broadway, but that might be harder to pin down.
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Liz
DD (3/2010)
"Make mistakes! Get messy!" - Miss Frizzle