Troubleshooting a broken speaker system
well, to be honest you'd have to have a decent understanding of the circuit to really track down a component failure.  

That being said i've fixed a few of my own internally amplifier speakers, it was almost always a solder joint that cracked or came free from its pad, probably as a result of thermal stress.  

So one thing you can do is look over all the solder joints carefully and re-solder any that look like they are possibly cracked.  All you should need is a soldering iron, some flux and some solder.

EDIT: One additional common point of failure in such electronics: electrolytic capacitors.  They are used a ton in audio electronics and have limited life spans, look for any bulging or exploded cans.  They are pretty easy to replace if this is the issue.