Dave, End game is spending more time with her. Perhaps racing. Karts would be an easy answer but the indoor karting in Sterling closed and the karting at Summit Point closed. If you know of another local option, please let me know. I have a Spec Miata I'm willing to let her drive in an AX or DE.

SCCA Autocross has a few karting classes that allow kids from 8-12 (JB), 12-18 (JA) and 15+ (KM).
DC region has had strong support for it- check out:
http://www.wdcr-scca.org/events/autocross

-Larry

I can't really spend the next couple of hours typing my thoughts on this but since the end game is spending time with your kid, you need to sit down and think about what the kid wants...not what you want. The worst thing you can do is force them to do something that they are not 100% committed to... especially since this hobby is expensive, and more importantly, the risks of injury or worse are out there. Think long and hard about that.
It was easy for me, Robbie grew up with and around cars. He has always wanted to drive and knows a great deal about cars from osmosis. He started out with motorcycles, then we spent many Sundays competing in karts at SPK and learning racecraft and car control. When he got his license we spent many hours autocrossing and learning how to manipulate a car on the edge of adhesion. We were fortunate in that I belong to a group that sets up and holds autox in Cumberland so on Friday afternoons, we go up to the airport to set up and have the whole course to ourselves. I waited until he was 17 to enroll him in the SCCA comp school (one of the best deals out there by the way and can start as early as 15) just to make sure the maturity was there to respect what getting behind the wheel and going 100+ mph really means. Signing the waiver that states that I give my permission for him to participate in an activity that can result in serious injury or even death weighed heavy on me for a while. But he did well in everything he did and if there ever would have been moment I saw something that he did that put him at risk, I would have taken him out right then and there. From there is was an easy transition into PCA.
But most importantly, realize that you are the dad and what you say may or may not be taken as gospel. I am fortunate to have been able to put Robbie in a car countless times with some of the best drivers I know and are people he respects and give him good advice. They have personally taken stock in him to make him the best he can be and he has absorbed this and applies this knowledge. My friends are his friends and he feels part of the group. This I feel contributes a lot to his success in motorsports. You need to make sure you have something similar. I just wish I had the resources to move him forward in his desire to be a professional driver. Right now I just want him to stay in college.....
I guess bottom line is find a group such as those mentioned above to get her into karts. See if she takes to it and has the skills necessary to be competitive. Make sure you have good equipment...won't last long if she finishes last all the time. Kids don't like to lose either. And be prepared and committed to spending more time that you originally thought.
Just my $0.02....

Pitt race ....bit of a drive but a lot of karting options

Some Inspiration
 

A year or so older but....
http://pbs.twimg.com/media/CJj_f0gWIAAXqWS.jpg
Carrie Schreiner just finished her first season in ADAC Formula 4 after running karts. Here's her website in case your daughters are interested: http://www.carrieschreiner.de/news/ (most browsers can translate somewhat from German)

Go to the World Karting web site. You can race almost every weekend. lots of seat time for less money. You learn racecraft not just car control.
They race at the Daytona Speedway between Christmas and New Year over a 1000 entry. They race on the same road courses that the cars do all over the U.S. . You can also run sprint races almost every weekend. Check them out. Also look at Woodbridge Kart Club.
At the Daytona race you get a lot of drivers from Europe.
Ed

WKA Daytona Speed week had over 3400 entry this year.

My cousin and his son are active in WKA and did not recommend it for beginners. He told me practice sessions were shortened to 4.5 minutes at Daytona. Just not enough seat time for beginners.