If you don't want to deal with the wiring, there are dozens of pre-assembled LED devices out there, some of them super cheap. You've just gotta get a little creative with re-purposing what you can find.
I've had pretty good luck with finger LEDs - bought a big pack of 'em off Amazon.
The most recent set I got had colored LEDs, but I've bought some in the past that are just plain white LEDs. They're compact and relatively bright, and last a pretty good long time. In some of them the batteries are changeable (but the batteries may cost more than a new one with fresh batteries). In some of them they're glued together and the batteries are not changeable; you just toss 'em out when the batteries are done, and put a new one in your prop.
I've also used the LED light assembly out of a candy/toy thing like this:
It's red (just right for Velvet's Psyphers), extremely bright (like, uncomfortable-to-look-at bright) and the batteries seem to last a long time. If you disassemble it, there's a little gobo/lens that makes the projector shape (a ghost or smiley face or whatever) - take out that piece and it's even brighter. Get one from the gas station or grocery store for a couple bucks.