M.N.
It's not $150 a night, but the Millenium off Broadway ( I think it's around 53rd)is nice and so is the Hilton ( same area,about 2-3 blocks south). I've personally stayed at the Millenium for business and a lot of the people I work with that go to NYC frequently stay at either of those hotels. We're on a budget too, so I know they're not more than $250 or so a night. Also, when I used to travel to NYC on a monthly basis with my previous company, we stayed at either the Marriot Times Square or the Marriot on 3rd Ave. Both are nice and again, since we had a business budget, they weren't more than $250 or so a night.
Broadway shows are awesome!!! You should definitely catch one while you're there. Wear comfortable shoes too, b/c if the weather is nice, it's much better & cheaper to walk everywhere than take a cab.
Have fun! You'll have a blast!!