Apart from working on the wires that are visible behind the belt, there's not much you can do as far as repairs go without significant and possibly damaging surgery on the belt's battery box or other parts. Everything is glued together so you'll have to cut plastic to get anything apart. The wiring is ultra simple and there's not much that can go wrong. For everything to fail, the issue would have to be at the battery box, otherwise at least one part should light up.
The easiest way that I can see for a problem is that the batteries are not inserted with the correct polarity or the small plastic tab has not been removed. Consult the small printed instruction sheet for battery placement, open the battery compartment and make sure the batteries are the right way, one up one down, and that there is no plastic tab preventing them from making contact. If I recall correctly, the battery closest to the switch is supposed to be positive side down and the one closest to the screw is negative side down.