My experience is pressing Power using a quick and short press. A slow and long press could lead to off and on again. Of course, hardware problem needs attention inside.
Maybe I can be of some help... I am french and I live in Vietnam (Saigon)
I cannot fix your problem but I have a good solution : I have myself a HP Jornada 720 that I NEVER use. That mean it look like a new one and work perfectly EXCEPT that the screen has died of old age due to humidity (mushroom??). So I can sell it to somebody who have also a Jornada 720 (or 728 I suppose) AND a screen if no problem. No problem to take some photos or making a video like you do.
If we agree on a price (maybe 30$ or 40$ less than the 140$ average price for this kind of machine), we could made a deal. You or the guy to whom you will sell your Jornada.