I have another TaoTronics product (the TT-SK03) which is a smaller round bluetooth shower speaker that charges via micro USB. It stopped working; it would charge but wouldn't turn on. TaoTronics took responsability and sent me a new one. The old, broken one I tossed in a drawer instead of throwing it out.
Today I found it and decided to open it up to look inside (and maybe find out what went wrong with it). It has some rubber plugs to waterproof the screws which I carefully pulled out with the point of a corkscrew and then I opened it up.
There is a speaker, a battery, and a tiny tiny control circuit with something like 4 wires coming off of it. All the electronics are covered with a waterproofing coating (like glue from a hot glue gun) and I noticed that one of the wires seemed to be moved a hair out of place (I don't even know if it is soldered or just glued into place). I pushed on the wire with a screwdriver to move it a little and now it works!!!
You may have a similar problem as the light goes on when you're charging, but it doesn't start.
Good luck fixing it if you still have it!