The W is walking distance to 30 or 40 dining choices from casual to fine. (Mostly casual as the area borders UCLA campus). New Years should be quiet in the area as the students are mostly gone, but you are a 5 minute drive from Bev Hills Restaurant Row and Rodeo Drive (or you can just hop the bus). It's not much more than 10 minutes to Santa Monica on the weekend, too....and you're only a few minutes from the spectacular Getty Center. If I were you, I'd try to get a dinner reservation at the Getty for Saturday night...a late afternoon visit to the museum followed by sunset cocktails and then dinner would be great. You'd then be able to fit in a play at the Geffen or a movie at one of the old-time movie palaces in Westwood. Or you could drive out to Malibu and enjoy a seaside dessert somewhere...Malibu is just a 15 min spin down Sunset Blvd from the Getty.

My step sister booked this hotel on hotwire:

4 Star, Beverly Hills, Westwood, Century City
113.00

It was the InterContinental and she LOVED it!
She said they were treated royally and she had such a blast.
I stayed here when it was a Hyatt and the location was fine. I don't have any problem driving everywhere but it's close enough to take cabs.

Looks like it might be 282.00 on hotwire.com per night during this time!

The W is a great hotel, much better than Century City.

There is so much to do downtown - I could spend a week there! All of this is within walking distance from you, some more than others:

The jewelry district- (roughly) along Hill Street between 5th and 7th is steps from your hotel- is the largest in the country. BONUS- it's located in historic theaters and department stores from the 1920s, so be sure to look up and enjoy the incredible architecture.

The Fashion District is huge, in easy walking distance from your hotel, and tons of steals to be had:
www.fashiondistrict.org

Grab a map for the DASH (the local downtown bus that costs $0.25) and scoot over to Chinatown for lunch or dinner (or breakfast, if you like chicken feet and dim sum!)
http://www.chinatownla.com/

Little Tokyo is very quaint and walkable, with many small, family owned shops that have been there for nearly 100 years sometimes:
http://www.visitlittletokyo.com/bid.html

here's a link to the big Business Improvement District in downtown, which covers your hotel area- I think you'll find a calendar in there for the big, grand events going on downtown:
http://downtownla.com/

The Los Angeles Conservancy does EXCELLENT walking tours every Saturday morning at 10 am, leaving from the Biltmore Hotel on Olive Street side (about $12/person and you have to book in advance, but worth it!)
http://laconservancy.org/

The Los Angeles Public Library is adjacent to your hotel, and worth a visit. It was all restored in the 1990s and is really impressive. Eat at Cafe Pinot outside in the patio/courtyard area- great food and great ambience.

For New Year's, I don't know your taste in music but there is a HUGE event that goes on in the streets of Downtown LA called Giant Club or something (used to be called Giant Village- not sure the new name for the NYE event). Tix are $80 and its lots of live music amidst the skyscrapers (or at least it was in 2005 when I went)
www.giantclub.com

There is also Staples Center (which has music events and sports events) but just opened adjacent to Staples Center is LA Live- a Times Square style adult entertainment (not in a dirty way!) district full of fun.

I could go on and on and on. If you have specific interests, let me know and I'll answer. I don't live in LA anymore (so forgive me if any of my info is old) but I LOVE L.A!