Yeah, the hairdryer works on the electrical tape. Thanks guys!
My other puppets all seem to have felt eyes. This looks just fine. I'm not sure which I prefer. It looks like most of the muppet show puppets, at least the non main characters, had felt eyes. Those felt pads for the bottom of furniture do work just fine and come in many sizes at Craft stores. I've actually taken to just cutting out the circle of felt and hot gluing it on, which works fine too.
Another note about eyes: I found some clear plastic bulbs at Michaels. They are in the Wedding section and I guess you're supposed to fill them with candy or something, as a wedding favor. Anyway they come in 2 pieces and there's a ring on one, to be used to tie ribbon or whatever to. This can always be cut off or hidden. Anyway, I intended to just paint them on the outside to make my eyes. What I did instead is sprayed the inside. This gave it a smooth white color still but kept the outside shiny, much shinier and glossier than the paint would have been. It probably grays it down just a bit since you're looking through the plastic at the paint, but the difference is negligable. Then I affixed the felt pad to the outside. Gave it a nice look. Not for every characte, but I'm making an elderly man so it kinda has that watery eye look that fits.