All the hotels in NYC that are cheap are actually still relatively expensive, and they are awful. Have you thought about staying in a hostel? That's where you'll get the best deal. And you'll meet interesting people there from around the world. The New York hostel is more expensive than most hostels in the United States but, under the circumstances, it is probably the best deal you can get. I've stayed in youth hostels fairly often, and I've always enjoyed them.
If you want more information about the hostel in New York, you can get it at:
https://www.hiusa.org/hostels/new-york/new-york/new-york-city
New York is a great city and it has fantastic art museums. I'm sure you have a wonderful time.