The price limit you suggest is unreal. Only a time machine taking you back to the 1960s would get you $100/night near the Theater District.
May is a good time, schools not out yet, weather usually beautiful, no holiday crowds unless you hit Memorial Day weekend.
Side streets off Times Square have some interesting ethnic restaurants. Taxis are very expensive and vulnerable to Manhattan's very slow moving traffic (recently measured at 6 mph!). In any case, subways and buses go virtually any place you might want to see, and are much less expensive, especially if you get a 7 day Metrocard.
Places to visit: Lower Manhattan--9/11 memorial, stock market, Chinatown (eat at Wo Hop), South Street Seaport. Also, from here you can get the boat that visits Ellis Island (immigration museum) and Statue of Liberty (same boat, one price, get off at either or both to explore and take a later boat onward, start early as long lines form)
Greenwich Village is good for nightlife. United Nations is free or you can take a paid tour. Gift shop very interesting for souvenirs from well over 100 countries
Central Park with zoo and John Lennon Memorial. West side of park has Museum of Natural History/Rose Space Center; east side has Metropolitan Museum of Art. Either museum could take a full day