@thegarse I could be barking up the wrong tree, I'm awake at the wrong hour - BUT... if you twist the power button at the back of the IIsi, it should recess and that will tell the PSU to always be on, bypassing soft power essentially. I don't know however if, without components on that card, that it will work properly still.
However...Why not just put the components that fell off back on? Which ones are you missing? Most of it is diodes and resistors, couple of caps and the main IC... you could just measure components and match them up against the schematics and refer back to a photo to make sure everything is in the right place.
Higher res photo mid way through cleaning/recap, for some reason I never took a photo after it was finished.
The top capacitor is a 100uf 6.3V cap. The bottom one next to the M51977FP is a 33uf 25V cap.
There are schematics for the AP41S sub-board / daughter board but I am finding them a bit hard to figure out (again, possibly because I'm awake at the wrong hour). I've attached the PDF that contains them, on pages 12/13.
It is a little tricky as I cannot remember if there's markings on the back of this board or not, but I think with enough effort we could figure out what goes where on this board.