This would be an easy choice for me. For a special birthday dinner, I would want someplace with a charming or elegant ambience that served food well above the norm in terms of both interest and quality.
Yongsusan on Vermont is the perfect place. It features Kaesong cuisine from the ancient North Korean capital of the Koryo Dynasty, which is renowned for its highly developed, sophicated cuisine. The food is delicious, interesting, and carefully prepared. Yongsusan offers many multicourse price fixed dinners, but the prices, especially considering the quality, amount, and complexity of the dishes, are a real bargain. If it is your first visit, I'd highly recommend choosing one of the more expensive menus. You'll get a bigger sample, and some wonderful dishes that, because of the cost of their ingredients, are excluded from the less expensive menus. After you've been there a few times, you can pick and choose among your favorites, or order off the a la carte menu. There are some very special and wonderful dishes here, such as the bosam-kimchi, made from a whole stuffed cabbage. It's amazing!
There are a large number of private rooms at Yongsusan and, depending on the number of people in your party, one of the private rooms may be just the thing. The main dining room features silk covered walls and Korean antiques. Korean classical music plays softly in the background. Even the waitresses are elegantly attired.
Yongsusan is one of my favorite restaurants in Los Angeles. If it were my birthday, I'd be thrilled to be taken there.
re: Tom Armitage
I agree with Tom's recommendation but add this caveat: if you are bringing a birthday cake or other surprise don't expect the staff will grasp your intent. I had delivered a birthday cake to the host on arrival saying it was to be a surprise but toward the end of the meal the waitresses entered, announced they were bringing out the birthday cake, cut off the lights and left us sitting there for a good two or three minutes.
That sounds petty though; the food and presentations are terrific and I'd choose it again.
One of my favorites is Chung Ki Wa (3545 W Olympic Blvd). Their BBQ is good as are their cold noodles (aka neng myun). There is a very good spicy fish dish as well.
Another restaurant that seems to get a lot of coverage in this board is Dong Il Jang, which I think will also work well.