Well, believe it or not, I was able to tin the broken ends of the microscopic copper wire. I took off the remaining ferrite bits from the solder pads, put some nice leaded solder, and was able to solder the tiny wire bits to the pads. There is good conductivity between the two pads through the inductor. Before it was completely severed.
But... the light still isn't working. It's still displaying the same symptoms (when I click it on, the led faintly blinks on for a split of a second then shuts off, not matter which mode or position it's at) Either the inductor is too damaged, or something else must be damaged that I can't see. The other much bigger inductor is a little cracked at the ears, but it's not sheared off like the tiny one was, and it has good conductivity too. Something else that I can't see must be damaged. I think.
Maybe a bigger capacitor hidden in the middle, or a transistor.
I don't know enough about this stuff to properly diagnose it, and my multimeter doesn't test inductance so that doesn't help lol. I'm not sure why it's behaving exactly the same now that I've reconnected the inductor, as it was doing before when it was completely severed off.
There is no value written on the intact side, it must have been written on the soldering side (there seems to be some very faint white marking left, unreadable now of course).