From the track connections underneath, I think the output IC is an STA540. This is a rather nice quad audio amplifier. Two channels are being used in bridge configuration for the woofer, and the other two are used in single-ended configuration for the two satellite speakers.
First check for the DC supply voltage across the big electrolytic. The STA540 expects 8~22V (it will probably be near the top end of that range, for maximum output power).
Assuming that's present, the IC has three control/diagnostic pins you can check before ordering a replacement (Digikey have them).
Pin 6 is called "SVR" and is connected through C30 to ground. It should probably have about half the main supply voltage on it, though this isn't entirely clear from the data sheet. If the voltage here is much less than half the main supply voltage, C30 might be damaged.
Pin 7 is the standby input. It should normally be driven with a voltage, typically 5V, through a current limiting resistor. Your photo seems to show it connected to the main positive supply, which would be a design error according to the data sheet. It's not clear whether the pin is connected to the supply rail, or not connected at all. If it's not connected, the data sheet doesn't say what the device will do. Can you measure the voltage on that pin?
Pin 10 is a diagnostic output pin. It's an open collector output that closes to ground if there's an output short or thermal shutdown occurs. You can measure it with a digital multimeter on resistance or diode range, with positive probe to pin 10 and negative probe to ground. If it indicates anything other than a high resistance or open circuit, the STA540 is telling you that there's something wrong.
The STA540 includes anti-thump circuitry so it should not make any noise when you turn it on and off. It might actually be working, but you can't hear any sound from it. You can try injecting a signal on pins 4 and 5 (which are connected together) (the signal should be audible on the woofer) or pins 11 and 12 (should be audible on the satellite speakers).
If the output IC is dead and gone, it should be worth replacing it, but check the other ICs too.
Can you tell us more about the nature of the "large surge"?
edit: Sparkfun sell a kit that uses the STA540, and they sell the IC separately too, for USD 2.95.