First, remove the sticky feet and the screws hidden beneath them. Then, open the case carefully. I opened the case, but not carefully, and as a result sent two small chunks of the power cord receptacle through the room such that I heard each of them whiz past my head twice. Neither was ever found.

The power supply lifts right out of its plastic case. Inside, there's a little square molded bracket for the fan, which will be lightly held in place by sticky stuff.

Power receptacle with broken-off pieces (not) visible.

The power supply, sans case.

A poor look at the damage. This is the cord from the supply to the laptop. It's got a core conductor (braided 16 gauge wire), an inner white jacket, a second surrounding conductor (woven wire, presumably 16-gauge-ish) and an exterior black jacket. It seems the interior jacket frayed through and the two wires shorted together. That was hot enough to vaporize a fair amount of wire and set fire to the exterior jacket. This is a fairly poor design, but it's the trade off you make to get a cheap, flexible cord. I think, but it's hard to say for sure, that the short happened at the junction between the black-jacketed conductor visible behind the flex arrestor and the surrounding conductor on the cord. The flex arrestor obviously did not do its job. A longer one would probably have helped.
The fan is a really good idea, particularly given my later findings, but is still kind of a joke. There's no roow for anything even approximating airflow in this thing.

I need to pull the fan, but the manufacturer likes epoxy, so I'll have to loosen the gob holding its connector to the header. It's not really clear whether this was intentional or just a stray blob.

The little teeny fan, shown here larger than life.

Next job is to remove the copper shield. I don't know what it is about power supply engineers, but they all love shielding. Probably my utter lack of fascination by shielding explains why I am not a power supply engineer. Anyway, it's pressure-fit in place and held there by conductive tape. Peeling the tape away very carefully...

Now we can get our first look at the guts of the unit. There's only one really big capacitor in there, the brown cylinder in the middle. I couldn't see what it was rated; if I were less lazy I could calculate it from the discharge time, but, well, "if I were less lazy" has a lot of other more interesting possibilities.

Four clasps fasten various components to the side. Honestly, I don't really know why. In one case, it appears to be to maintain a good connection, but the three have a little insulating blanket that completely engulfs the components, so it's less clear. I blame shielding. As you can see, removing the clasps requires the use of a special clasp-removing tool. There are also three screws holding the board to the frame. What a great time to remove those too.

With the clasps out of the way, the shiny metal frame (which is for shielding) comes off revealing, yes, more shielding.

Fortunately the insulation comes right off. The various grey bits visible are all that's left of the shielding.

Here's a look at the bottom of the board. I've drawn a white circle around the white (negative) wire's solder point, and a black circle around the black (positive) wire's solder point. As you can see by the precision of my circles, art represents yet another career path that is clearly not for me.

Here is a close up of the damage that nevertheless manages not to show any more detail. I really hate taking flash photos of shiny things.

Here you can sort of see streaks of white where the inner jacket melted into the outer one.

I peeled the outer jacket back to see what was going on. See all those short wires hanging down? Those are the ones that vaporized. It's a huge mess, so there's nothing left to do but cut the wire. I'll pull the flex arrestor off because I'll want to use it later (it fits the hole), but I'll push the ferrite bead up the wire a little bit because if it comes off it'll be a pain to get back on and, you know, they're good for shielding.

I'm going to pull the heat shrink tubing out of the flex arrestor so I can use it again (the arrestor, not the tubing).

That leaves us with a bare power supply. Time to find out if it still works!