You will need to identify which wire is which (a simple multimeter on continuity test should allow you to find the ground pin) and then arrange a circuit that will connect the signal wire to ground when you need to. It sounds like the button is an active-low connection - connects a signal wire to ground when you press the button. Probably because the capacitor is discharged when you attach it, but from then on it's charged up. That was why battery led is flashing when I solder the capacitor to the pins of the power switch.
What I did causes 'pressing button as soon as the AC power comes' effect.
The laptop expects power button released in the firs place, and then pressed for a short time. If I solder this capacitor in the same place, the laptop is not getting up when AC power come back. If I touch a 1 uF capacitor by parallel, laptop powers on. Unlike desktop computers' BIOS, most laptop's BIOS including mine does not have such power settings and no 'wake on lan' option.
The problem is, when AC power lost and come back, the laptop can not be powered on automatically. When I need a hard reset, I want to be able make this happen with a toggle of AC power. I'm running the laptop without battery in case of a need for a hard reset. I'm using one of my old laptops as a server.