To house the coil, I got an old picture frame from my local Goodwill. I removed the picture and painted the back of the glass with silver mirror-effect spray paint. 

It turns out, this paint is slightly conductive! It must have real metal flakes in it. I attached the coil directly to the painted surface with clear packing tape, and it stopped working. The metal in the paint was enough to disrupt the coil's magic from doing it's thing.

I removed the coil and put a layer of clear packing tape down on the back of the mirror, then re-added the coil with another layer of packing tape to secure it. This fixed the problem.