After hours of tracing cables, removing panels and taking apart more of the car than I'd like to admit, I figured the issue out. Apparently there's a cable for control current (or something like that). The control current had gone out. The car still kept thinking the lights were on, as the LED control circuitry was apparently using the power the car was feeding it to illuminate the LEDs it wasn't illuminating. That's why there was not indication on the dash about the failed lights (the indication came on once I disconnected the power cable to the lights).
I traced what I believe to be the control cable back to close to the battery from a mysterious small black box next to the right headlight, where I found a casing for a glass fuse on the wire! To no one's surprise, the fuse had blown. Having now replaced the fuse, the lights are back to working as they should.
For the record, I do feel like an idiot. :D
In my defence, the additional fuse casing was sort of hidden behind something else in the engine compartment. Obviously it is related to the aftermarket headlights installed on the car by a previous owner.
So in summary:
- check all the fuses
- make sure you checked all the fuses
- verify there are no fuses you haven't checked
- locate the extra fuses you still haven't checked added with aftermarket components that are not in one of the car's fuse boxes and replace as necessary