At 12V the LEDs likely would burn out. From what I've seen looking at LEDs in electronics parts stores, mostly they are rated to be powered at 5V or a bit less. The small resistor that Antec includes in the LED wiring (concealed beneath the short length of black heat-shrink tubing just behind the Sonata's front bezel) is there to protect the LEDs against excessive current. Antec calculated the resistor value assuming 5V applied to the LEDs--at 12V you are more than doubling that applied voltage, and therefore would be forcing an excess of current through the LEDs (at least in theory, and according to my rather limited understanding of electronic circuit principles). I suppose you could satisfy your curiosity by just going ahead and feeding 12V to your Sonata's LEDs to see what would happen. If you hear a "pffft!" sound, then you'll know with certainty that 12V isn't a good idea.
Good news is you can always get new LEDs from Radio Shack or any other electronic supply house, although the blue ones can sometimes be more expensive than other colors (depends upon your supplier, I guess). All in all, adding in the little trimmer pot is really quite simple, especially if you can do an adequate job soldering. If you don't have a solder iron, then you could probably make good connections using small size crimp connectors and pliers. The trimmer pot itself is tiny enough to actually glue it to the surface of the molex connector attached to the Sonata's LED wires. That way the trimmer wont be dangling loose inside your case, and you'll only need very short lengths of wire to make its connections to the molex.