I think getting your daughter a dog is a great idea. I have a friend whose parents got her and her two sisters all a pet of their own when they turned 13. She got a corgi and it was the love of her life.
I work at a boarding facility and a customer of ours is a lady who has a young teenage daughter who was very shy and very depressed. She got her daughter a pomeranian and it helped her daughter a lot. She is still a very shy girl, but the depression is a lot better. The little dog gave her something to love and care for, a reason to get up and be happy. The woman said that her daughter takes care of her dog just like a new mother would care for her baby. The girl even rocks the dog to sleep at night.

Bitsy, My pom is very mellow. But oh the grooming. Poodles are a great suggestion, but also can be a lot of grooming. I wouldn't recommend a yorkie (hard to train), jack russell (hyper), or a pug (hyper). I don't think a spitz is the best idea either, they are a lot of grooming and I've seen many aggressive spitz. Not that all are aggressive, but at the boarding facility I work at the majority of the spitz that have boarded with us are aggressive and not allowed back (seriously, I would say 75%).

I think mutts are great. But as far as breeds go, corgis are usually dogs that get along nicely with other dogs, also dachshunds. A little more active, but a breed I like a lot is rat terriers. Papillions are great too, kind of similar to a pomeranian but a little less grooming.