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Rates are low for these dates (tbh, i'm somewhat surprised given the length of your stay) but depending upon how flexible you are in terms of zones, these are my thoughts...
If you're flexible on zones i would suggest taking this Priceline Express offer (w/ the extra 5% discount offered thru the board's Priceline Coupon):
4.5* Madison Square Garden - Convention Area (<-- CLICK HERE and confirm your dates on the landing page) for $150/nite with:
Free Internet in room
Free Internet in public areas
==> best guess for this property would be the Crowne Plaza HY36 Midtown Manhattan which is a fairly new hotel, gets excellent reviews, and is centrally located a couple of blocks from Penn Station for easy access to multiple subway lines. Not sure how you feel about this area, but it's just south of Hell's Kitchen and north of Greenwich Village, both of which you mentioned as being desirable (so i would think this would offer good access to both).
Could you possibly bid on PRICELINE and get this for slightly less, but i wouldn't necessary want to take the risk of potentially landing a different 4.5* property in the zone in hope of saving a few dollars.
If you're 'set' on the zones above, we can suggest a bidding strategy for you to follow, and you should be successful at under $175/nite, but again, you would need to willing to accept any 4.5 star (or higher) rated hotel in the zones that you're bidding, which puts something like Kimpton Ink48 Hotel New York into play (ie- a 4.5* hotel in the Hell's Kitchen zone) which gets very good reviews, but not exactly the most central of locations.
Please remember to return to BetterBidding and use the PRICELINE and HOTWIRE links on the board to begin your travel purchases... or thru our sister site MAPSandHOTELS.com for your regular rate purchases.
Let us know your thoughts on the above (or how it goes if you take the guessed Crowne Plaza HY36 Midtown Manhattan offer above) and how you'd like to proceed, and we'll take it from there.
PS- if you're new to priceline bidding, or if you bid often, you may notice that the Name Your Own Price is becoming more difficult to find as priceline is constantly moving/hiding it, we have therefore created the following page for easy access which you can bookmark and/or use: BiddingForHotelRooms.com