Yeah, I like that idea too!
Here is the highest Farad capacitor (200mF) I could find with a Z-height that was remotely possible for implantation (1.4mm). If the LED is running at ~3V, the math for how long it would last would look like this:
Farad = Coulombs/Volt
(200mF) = Coulombs/(3V)
Coulombs = 0.6
Amp-hours = Coulombs/Seconds in an Hour
Amp-hours = (0.6)/(3600)
Amp-hours = ~167μAh
If a tiny LED required 15mA to reach full brightness, that means it would last:
167μAh/15mA = 0.011h = ~40 seconds
That’s a pretty big implant for 40 seconds of illumination. Some other biohackers I know are working on developing their own high density Supercapacitor to address this problem, but it’s not exactly viable as an off-the-shelf solution at the moment. We’re pretty much stuck with chemical batteries or magnetic field power.