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I don't have an X360, but I have a 10 year old daughter and a Wii, so I probably have a pretty good idea of your dilemma. In fact the reason why I don't have an X360 is because of her, there were just more appropriate game options on the Wii. A lot of games that fall under the "family friendly" grouping are, quite frankly, terrible. It doesn't sound like that is what you are after anyway. It sounds like you are looking for games that are actually good that happen to have a female protagonist. Since I know the Wii, that's what I'll stick with.
With a Wii, it sounds like the Harvest Moon series might be something she would enjoy. There are three different Harvest Moons on the Wii, Magical Melody, Animal Parade and Tree of Tranquility. All but Magical Melody allow the player to choose either a male or female character. The basic Harvest Moon formula is that the character moves into an old farm that has seen better days, and rebuilds it to its former glory. These are long, relatively slow moving games, but if she likes Animal Crossing, she will probably like Harvest Moon games. These games are much, much more fun that they sound like they would be. The main character can also get married and have children, which is pretty unusual in a video game. A spin off series is Rune Factory, which adds a fighting element, and one of those are available on the PS3, but I don't think they are on the 360.
One series that definitely portrays a strong female protagonist is the Metroid series. It may be a little too intense for a 9 year old, since there is a lot of blasting aliens, but my daughter likes to watch me play them and I'm ok with her watching. They are rated T though. Metroid Other M may be a better choice than Prime, because it is story driven and has many cinematic cut scenes, but some of the boss battles are pretty challenging. Still, for a strong female character that is the opposite of the helpless princess, I don't think Samus Aran can be beat.
Sorry didn't mean to write an essay. There are other games, I'm sure, but these two series are our favorite "girl" games on the Wii. There are also many games that have nothing to do with gender at all. World of Goo is excellent, and downloadable on the Wii as well as Smartphones. Perhaps someone else can suggest some X360 titles.
This link to a page on "great games and websites for can-do girls" at Common Sense Media might be helpful. It lists and describes a bunch of games, broken down by age groups. I counted eight in the 7-8 group and another nine in the 9-11 category.