(Yes, I know this is a Christmas card, but I'm working on ideas for a card swap.)
Using the 1/8" punch, make a hole in the desired spot.
Insert the eyelet in the hole and using the 1/16" punch, put the shaft of the punch through the long part (the back) of the eyelet. Squeeze and voila! You have and easy set eyelet. This won't work beyond the outer edges of whatever you are making, but it's great when you can use it (and a lot quieter.)