Take out your Bare Paint Pen and put a few drops of paint on top of the legs of the LED. You want to electrically connect the legs of the LED to the surface of the badge. Feel free to use as much as you'd like, but make sure that your blobs don't run together and touch or you'll short out the LED. Also put some blobs on the battery holes, this will make things easier later.
Notice that I was careful to make sure they weren't touching in the middle.