A couple of days later I went back to it and tried sanding the wire to get better contact. To my surprise inside that thin wire is really thin copper wire. It was working so I soldered the wires together and ran the cord down the handle and inside of the box. I used hook and loop to mount the battery box under the seat so that I could easily remove it for battery replacement.
If I was going to do it again I would have gone with a soft white LED
The end result is great for cars being able to see us, the kids have light, and I'm happy with it. I realize this is pretty basic but for a guy who is terrible with anything electrical it was quite satisfying.