On the first pic is an assortment of some of the parts involved. Despite the loud report of the power supply's failure, the hole at the middle of the cap (upright position near middle) is barely visible and easy to miss, although the slight outward bend of the top plate is more reliable indicator of a failed cap. After removal, from the leaked and burned electrolyte at the bottom in the second picture, the failure becomes much more obvious. It's a 10uF 450v 105C piece, about 21x13 millimeters.
Finding a suitable replacement part is often an artform of its own, because people don't usually keep components of every possible value and size hanging around. On the far right is the first cap I tried, which I had to solder dangerous extra leads to because it was snap-in type and so big it wouldn't fit in the original location at all. The 33uF one at the bottom is the replacement I settled on in the end, in part to avoid another blown cap, although it required cutting a bit of plastic off the insides of the cover.
At the top is also visible another nearby capacitor in the SMPS section (47uF 25V 105C); it doesn't even read on my effective series resistance meter, suggesting it's totally dried out, although there's no outward sign of failure. I should probably go through every cap on the low voltage side to make sure they haven't failed as well. The feedback opto-isolator didn't show any outward damage and the diodes checked out fine, but I replaced it just in case because I didn't want to set up a test-bed to see if it worked. For the actual switching IC there were no indications of how it should measure if it's intact, but Google searches suggested it would be almost always bad so I replaced it as well. One with short leads on the left is the original, one on right is one of the replacement parts.
The blown fuse was a matter of its own. Being enclosed, there was no indication of failure on the outside, but of course being it's a fuse, it's easy to measure even in-circuit. It wasn't until I cut it open that I realized it should probably normally be possible to screw the top open (though I wouldn't recommend this as it might influence its integrity). As can be seen, this one is quite clearly blown.
I already threw out the bad rectified bridge, but as usual, there was no outwards sign it was bad either, but again due to the nature of it's function, it should check out rather easily even in-circuit. The schematic I've added is from the TOPSwitch II switching IC datasheet, it's not exactly same as the batter charger, but close enough.
It's probably worth reminding, especially if there's a fault in the circuitry, those capacitors can retain charge far after the power has been disconnected, and need to be bled empty through a power resistor before touching anything. Incidentally, with a shorted rectifier bridge like this, it seems like you could get a shock from the power-plug if the cap and fuse are replaced!