Interesting in general, and now I also get curious about how this works out.
I don't really have much good advice, I seem to be an aspie some people like to approach, but I don't know how to deal with the attention, and the connections, I don't know how to balance their interest, with my shortspanend attention and lack of social skills, and lack of maintenance of social connections.
Sometimes it is all right, to be silent for a while, when you run out of something to say. I know that is an awful feeling, but as a student in socal work, I was told that I should give other people a break, be quiet for a while, and don't panic with a desperate need to fill the silent "gap", but let others get a chance to think, and come up with something. The conversation is not totally your responsibility, so to make a dialog, you have to give her a chance to come up with something. And then follow her lead, of the conversation.
Listening is also a very good skill, not my strenght, but according to the books, you should be able to pick up something from what she says, and give it back with a question. In a way let her know you listen, and also that you show interest and want to know more about her and what she says.
Good luck! 