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Since you will be traveling light, you can easily get from LGA to Manhattan by public transportation for $2.25. Whether you can find a hotel in Manhattan within your price range is another matter. You can check the NYC lodging specials at travelzoo.com; you can see what's offered at hotwire.com; or you can submit a bid at priceline.com, after reviewing winning-bid information at biddingfortravel.com and/or betterbidding.com.
For sightseeing in Manhattan, try to stay in Manhattan. If you must stay out in Queens, try to stay at a property with convenient subway access, like the Holiday Inn in Long Island City.