NYC hotel rooms are notoriously small, so you have that going against you. The Marriott Marquis in Times Square has some of the largest rooms in NYC, and most include a sleeper sofa. But the price is generally more than double your budget (I didn't check these dates specifically). To save even more money consider looking at hotels that include a kitchen, such as the Residence Inn or Homewood Suites or Home2Suites chains or something similar, so that you can avoid eating out all the time.

I think anything that you find in Manhattan for that cheap at any time of year will most likely be a place you wouldn't want to stay. If that's your budget, you should definitely stay outside of Manhattan, either in NJ, Brooklyn, Queens, Bronx or perhaps Westchester to the north. But then you're looking at anywhere from 30-120 minutes to commute to/from Manhattan, plus the additional cost per-person for the travel that now raises your daily cost.

If you're willing to deal with the long commute, then consider staying at an airport hotel near Newark Airport or JFK. You will benefit from shuttle service to/from the airport, and decent mass transit options to get almost anywhere you want. For example, the Springhill Suites Newark Liberty International Airport is available for your dates for around $190/night for a room with two double beds and a sleeper sofa (no kitchen, but you get a mini fridge and a microwave). Or Home2Suites by Hilton Newark Airport for around $180/night for a room with two queen beds and a sleeper sofa and a kitchen.