Circuit Failure How to fix it?
Ok, so first of all I can't see things about that charger that might have safety implications so be careful, probe outputs for dangerous voltages, and assume anything you haven't measured that is conductive is dangerous to touch (even and especially heat sinks).

There are two capacitors on that board that may have failed especially if it's an older part. They are the cannisters on the left. Inspect them closely for damage, and if one is damaged you can try replacing it with a similar part (same capacitance and voltage rating)

Like the comments say, that's a switching regulator. Energy is allowed to flow into the inductor from a high voltage low current source, and flows out to a low voltage high current destination.

The 8-pin circuit in the middle appears to be a controller, and if you want a better understanding of the circuit find the documentation for that chip.