Yeah, that's on the southern end of Chelsea, closer to the Meatpacking District but not too far from the West Village.
Scarpetta, Morimoto, Del Posto, and Colicchio & Sons are all around there but probably more than you want to spend. It's going to be hard to get somewhere relatively quiet AND not too expensive.
Maybe the Tap Room at Colicchio & Sons, not sure how they handle big groups.
Legend - Two level Chinese restaurant, larger groups can probably do a semi-private alcove (with karaoke machine!) on the basement level. They take reservations, regularly do big groups, have a liquor license, and are relatively friendly. Avoid the Vietnamese dishes on the menu. Stick to the Sichuan ones like dan dan noodles, wontons in red oil, etc.
Pastis - Balthazar-lite. French bistro, something for everyone, but a little crowded and noisy. However, they do have a long communal table that may work with your group. The food isn't amazing but it's dependable.
Crispo - They'll ask you to do a prix fixe dependent upon what day and how many people, 10 would be a different prix fixe package than 14+ which is served family style according to their site. And it can be noisy, but I'm wondering if you could work with them regarding exact placement of your table, to try to find a less noisy part of the room.
A little further away:
Txikito - Excellent Basque tapas, with a variety of daily specials such as roasted fish, housemade sausage, suckling pig. They have a private room that might work.
Tipsy Parson - Southern, I think they'd require a prix fixe.
Salinas - Tapas, more Catalan influenced, I believe.
Co aka Company - Jim Lahey of "no knead bread" fame's pizza place, they only do large parties before 6:30pm though.