I am not sure that I have the same professional experience as the other commenters here, but I have been practicing various arts for 24 years, and teaching for about 14 of those.
My simplistic recommendation - put her in a class and with other students and teachers/senseis that she loves, and will continue to attend merely for the social aspect of it. No matter what sort of expertise she might develop in one style or another - its pretty much a moot point if she doesn't enjoy and internalize the lessons.
When I am dealing with children, my primary goal is to entertain and keep them coming back (and btw, I am not a professional teacher but a volunteer - so no argument that I am just extending their parents paying me to do so.) I want those kids to have a positive experience, learn some discipline, practice going beyond their mental capabilities (not in gung ho fashion, but learning the joy of mastering a challenge they had previously thought impossible.)
I wholeheartedly agree that TKD has done some fabulous work on a global scale in deliberately aiming towards younger practitioners. I started with judo and TKD at the local YMCA 24 years ago, and it instilled a love of martial arts in me, mainly because I was hanging with my buddies, and the teacher was super cool. I have never used those arts directly, but they certainly gave me a solid foundation.
In terms of specifics, when we are discussing smaller kids, I would think that the main lesson for self defense is situational awareness, and E&E if the threat demands it. This doesn't require a dojo, it requires hands on but non-hovering parents. Honey, if you feel threatened, here is how to get away, here is why you are right to be cautious (without scaring the %$^% out of her) and here is an example of my response to your actions (a big hug and a kiddie size AAR.) Show me the biggest 9 year old, with 9 years of martial arts training, and I am still going to be able to sit on the kid as a bad guy.
The above paragraph aside, there are one or two non-traditional modern arts, which would give a 9 year old the capability to injure. I would not (and do not for my nieces and nephews in that age range) advocate teaching them to children as the cognitive ability to delineate when appropriate to deploy those tactics is questionable. If we were approaching TSHTF situation in which I deemed the general public a threat as opposed to isolated BGs, I might consider doing so. Much as you folks and I feel this world is spiraling down, even living in downtown Chicago, I do not teach 9 year olds how to debilitate adults by taking out knees, collapsing trachea, and eye gouging. I read the Chicago news, the cop blogs, am aware of predators - but am still hesitant to teach kids these things. I dunno - I am an uncle and not a dad - perhaps that might be the decisive factor. I DID however teach some of these things to my sister and brother in law, if they choose to train their kids that way as parents, that is their choice... I will just continue buying the kids ice cream when mom and dad say "No sugar..." 'Cause thats the sort of crazy uncle I am proud to be.