A few notes based on when I did it(*):

You only need to remove the 6 outer screws off the bottom of the EEE, no need to remove the panel (or the motherboard later on).

When removing the top cover the audio sockets act as an anchor.

I didn’t remove the motherboard.

The two left-hand pins of the power switch are connected and grounded. The two right-hand pins are connected as well. The third pin at the bottom is connected to the switch chassis and also grounded.
Connecting either of the two right-hand pins to ground will turn the EEE on/off. I ended up with one wire soldered to the bottom-right pin and one to the bottom-center pin.

(*)I _mostly_ did it, I got some help with the soldering

If you’re still looking at a 12V-9V supply I’ve got a circuit based on a 7801 LDO (low-drop-out) regulator which is working well for me. I’d be happy to forward the schematic.