From my experience LED lights are not visible with NVD. Like Dave said, get some IR LEDs if you want them to work for you and research how to wire them properly otherwise it will be a waste of time and money.
Since you want to keep it cheap, take that $20 and buy a nice bright light (NO LED unless IR), then crack open a floppy disk and pull out the film and tear out the cloth. When you tear out the cloth try and get it in one piece (you will use this to wipe finger prints off the film). Separate the film from the metal disk in the center without ruining the film.
Depending on the size of the bezel on your light, you may need to open a couple more disks. Note that you can also use developed film, near the end of the color film negative strip there should be a
dark area, that is what you use.
Center the film under your light and trace it with a marker and cut it out. You will want at least two layers of film over the bezel, and if the bezel is too large you will need to piece the film together. If the light bulb is still too bright for your needs place another layer of film on it. Just note that adding a filter (VL filter) will significantly reduce output.
It does work and rather well for what you pay for it. Give it a shot, if nothing else you will learn more about IR/NVD.