Agreed with the Streeteasy recommendation. Not everything is on there, but there's more on there than anywhere else and it will definitely give a good sense of pricing in any given neighborhood (if you see outliers, assume there's a very good reason for those outliers, as it's extremely rare for a listed apartment in NYC to be significantly cheaper than true comparables). Most brokers in NYC are indeed sharks, although if you can get your work to pay for one, they can sometimes be helpful, particularly if you're not looking at big full-service buildings that have their own no-fee sales staff (and if your budget is $2k, you're probably not looking at big full service buildings).
I think it'll be very difficult to find a solo home in Manhattan for $2k or less. A couple of my colleagues have kids who just finished college and they're paying close to $2k each for tiny bedrooms in shared apartments with roommates (which IMO is insane). And studio sublets in my co-op (which is generally cheaper than the surrounding neighborhood due to a burdensome application and board interview process) go for $2300-2500. If you get out into Queens or the less popular neighborhoods of Brooklyn $2k becomes feasible, though.
Note that if you ARE looking at co-op sublets, make sure you fully understand all the rules of that particular co-op. They don't just require more paperwork, but often require an up front nonrefundable application fee that can run into the hundreds of dollars, and you'll usually need to be interviewed by the co-op board, who can turn down your application (and not refund your fee) without giving you a reason. Many co-ops also have restrictions on how long sublets can last (often 2 years, although not always).