I’m off to work here shortly, but I’ll give you a few more references and a bit more explanation. (Also, now knowing that this is for a one-off piece of jewelry, this might be less appealing a solution.)
Micro is micro-controller (like the chip that powers an Arduino as a artist reference). My core idea was “take a chip from the same broad family as the Arduino, but pick a tiny one, and build a custom printed circuit board (PCB)”
That micro has built-in processing for a touch sensor, meaning you could place a spot on the PCB that the user could touch, electrically connect that (as part of the PCB) to a touch pin on the micro and use that to turn the LED on/off. That way, your switch takes 0 parts and takes up 0 volume.
Use that same PCB to provide placement, wiring, and place for the resistance needed for the LEDs. Buy bare LEDs (0603 size is manageable by most by hand, with 0805 being easier but for a ring, pick one that looks good size-wise.)
To the points the others are making about the battery supply. Voltage is like electrical “pressure”. Pressure isn’t the only measure that’s relevant. I could have 1000’ of 1/4” pipe feeding a water spigot from the same source water as another spigot fed from 5’ of 3/4” pipe. If you measured the pressure of each with no water flowing, they would be the same. If you hooked a garden hose to each and tried to run a sprinkler on them, they’d work quite differently because the pressure would fall off a lot on the one with high internal resistance to flow (the 1000’ of 1/4” pipe, or the resistance inside the batteries you’re looking at).