SA-

You have some great choices in that price range. I'll give you a few suggestions below. Please review my recommendations with the reviews on http://metromix.chicagotribune.com/dining/ to make sure the place sounds to be your fancy. My recommendions are lower key with a more local flair away from the touristy spots.

With in Walking distance of the Drake:
The Saloon Steakhouse.. 200 E. Chestnut ST (Steak & more Steak)

$5~8 cab ride
Thyme.. 464 N Halsted ST (Modern Cusine)
Bistrot Margot.. 1437 N Wells (My fav authentic French bistro)
Adobo Grill.. 1610 N Wells (Upscale Mexican) You probably don't want to go there for NYE but you may want to have a dinner there during your stay.

$9~15 cab ride
Spring.. 2039 W. North Ave (Modern Cusine) also easilly accessible by CTA blue line train.

Avoid the Signiture Room at the 95th (John Hancock Tower) for dinner, but the bar is a nice place to have a cocktail if the weather is clear. If there is somewhere else you find and want some advise, drop me a PM. Chances are I have eaten there or at minimum heard about it.