There are a few strings of LEDs on a circuit board that are interleaved with the current to each one controled by a limit resistor near the connector on the led board. If the resistor fails, or of the alternator over voltages the LED's can get burned out. Sometimes the resistor fails and the LED's are fine.
The LED's are mounted on a circuit board and are very small. trying to move them would be possible, but complex.
Getting them out of the taillight basically requires cutting out the back of the lamp to extract the reflector assembly that contains the leds. It's a core and you loose material when you cut it out. The back is reflective as well, so if you try to re-assemble it you can see the cuts through the back. Thus you need to figure out a trick to re-apply the reflective backing. It's really quite a challenge to do it in a way that isn't leaving it looking worse.