First off - when an electrolytic capacitor dies, it usually blows out the top, where those score marks are (BTW - what's the brand name on the caps? Creative is notorious for using certain cheap brands of caps that die quickly). Even so, does the black goo seem tacky, or stiff? If it was anything from the capacitor, it would likely be a little runny. Also - what does the board smell like (are you aware of what normal PCB versus overheated PCB and parts smell like?). Sometimes that will give a clue.
That part in the back, mounted to the heatsink between the caps - that's a multiwatt15 (package style - same as an L298) audio amplifier of some sort. Can you read any numbers off of it? That would be one good place to start, if you can find a datasheet for it (I may have such a datasheet myself; I have a few such audio amps in a pile of parts I have that I found datasheets for - not saying it will match yours, though). Anyhow, once you had the datasheet, and know which pins are input and which are output, you could then probe with an o-scope (or maybe a small speaker - but a scope would be best) to determine if its getting the signal properly and amplifying it.
It could even be the control box. If you go to the Electro-Tech forums (http://www.electro-tech-online.com/), you might find a thread or two there on similar such Creative speaker system issue being discussed (I've seen them come up before).
Be aware, though, that while you may get good help there, its not as friendly an environment as the Arduino forums; I'm not sure why that is, but I suspect it is populated by more older engineers and such, and less artsy people, coupled with a different mix of international parties than here (it seems the Arduino forums have more Europeans, while Electro-Tech gets more South African, Chinese, and Middle-Easterners, who seem to have less "PC" views of the world - I could be wrong in my views, though). Still, they can be a helpful bunch, if you're patient.
You're probably also going to need to reverse-engineer the schematic on that amplifier board, once you have a datasheet for that amplifier IC. You'll want to start from it, and work your way out from it (drawing the input and output stages before and after than part). Whatever died may have taken out more components that just whatever is initially obvious; its possible the caps died in some manner and took out the amp, or vice-versa. Perhaps other components also died.
Finally - you have to ask yourself if all of this is really worth it; you might learn a lot, but you might end up spending more time and money doing so that you would have had you bought a new system. Only you can decide, though, whether that's a worthwhile tradeoff...
Good luck.
I just assumed that that was the problem, but then I realized maybe it's not. (I guess it still could be if all the specs don't match exactly?)
