When I lived in DC, I jogged in Rock Creek Park. It was next to my neighborhood. It’s a beautiful shady tree park, but a lot of normal joggers and bikers go there. :)
If she was doing business in DC and she exercises, she would likely go there to run.
That's my take. She was an avid runner. You don't want to drive to a park with your purse and leave it in the car. So you take your license, your car keys, and some water with you in the car. That's all. You don't leave anything in your car in D.C. or it will get broken into.
So when she was running, her license fell out. I read that she often runs up to 6 miles, so if she didn't realize until later that she lost it, that would be a lot of ground to retrace to find it.
She probably just reported it missing and went to get a new one.
It is beautiful. I stay near it a couple times a year. I always think of Chandra Levy, and picture scenes from movies where to politicos meet to talk off the record.