What kind of food do you like? Are you dining alone? Are you only interested in places that take reservations? What do you mean by "without breaking the bank"? How much money are you willing to spend, including tax, tip, any drinks/wine? What neighborhoods are you interested in? What cuisines do you like? Your query is very vague.
By Gramercy, you mean Gramercy Tavern? What do you mean by Bar Daniel? Bar Bouloud?
It is Restaurant Week during the time period you are here. Mondays through Fridays, prix fixe deals at many nice restaurants. Most of the most popular do only lunch. I just ate at Gramercy Tavern for lunch, they did a very good three course prix fixe lunch for $24.07. I would also recommend Eleven Madison Park for lunch, or the Bar Room at the Modern for dinner.