Have you narrowed the problem down to just the LED strip for certain ?
Can you substitute just the LED section for the other side of the car, or connect to the wires for the other side ?
The actual LED bulbs themselves almost never "burn out". When you see LED bulbs out, like in traffic signals, that's because of bad soldering, that's why entire strips go out.

Maybe you can make sure it's not some other part there, or the wiring connections. Did you follow the instruction sheet that came with the kit ? A PDF version of it is available if you did not have instructions. I see on the PDF, there is some sort of assembly you mount behind the seat, I assume some sort of voltage converter. I know you said it worked at first, maybe part of that stopped working.
I think if you can hook the sill up to where the opposite side of the car is connected, you can confirm it really is the LED strip itself or not. Then maybe the strip could be repaired if it is not all sealed in plastic.
Just trying to help troubleshoot.

Is the strip actually LEDs or is it an Electroluminescent strip ? Does it have discreet bulbs that make the panel strip light, or does it seem the entire strip lights up itself.
Now that I think about it some more, I'm guessing it is electroluminescent.
If it's not LEDs, an Electroluminescent replacement strip should not be expensive, here's an example strip I found, there's a lot of them available in different colors at various sellers.
https://www.adafruit.com/product/415...FQEHaQod0ToL-g

Maybe the unit you mount behind the seats is an inverter for electroluminescent strips, and that's how this was made.
That same site also sells inverters powered off 12V DC so if that is what you have, and that is what went bad, and the strip is still good, you could try a new inverter.
https://www.adafruit.com/product/448

Or start from scratch, using the stuff they sell and change to electroluminescent strips in a color that matches your new preference.