If you look at the EL Badge, it will also a made up of two circles. The front circle, and a back circle. You will want to cut a whole in the front of the hat just slightly smaller than the back circle of the EL Badge. Again if you cut the hole just right the EL Badge will be can be pushed through the whole and held into place without any adhesive.
Once you have the hole cut. Add adhesive around the side of the back circle of the EL Badge and insert it into the hole. Allow adhesive to dry.
Note: Though it may look more aesthetically pleasing to center the badge vertically with the hat, I would recommend putting the badge higher towards the top of the hat. If you have a big head like me, you won't be able to pull the hat down over your head further than where the EL Badge is poking through.