Replying here many months late but I think this is the same thing that happened to my Mom's computer. She past away a year ago and now only my Dad uses it (mostly for e-mail). He noticed the monitor wouldn't bring up the display anymore a few days ago so called me for help about it. I thought the power supply had failed.
Went over to see it and first thing I saw when I opened the side panel was a small aluminum heatsink sitting on the bottom of the case. A wire runs through the middle and acts as a spring which is supposed to be held onto the mainboard by what looks like circuit jumpers with a hoop on top.
One of those was missing, then found after turning the case over and shaking some.
I was a little surprised to find this message here at bleepingcomputer.com when I searched for anything like this ever happening. Must be a manufacturing defect.
I put super glue onto the clip and pushed it back into the "CLIP2" holes where it had fallen out. Not long afterward I had the heatsink back in place, but before I did that I bent the wire spring to make it have less strain on the clips. No thermal paste available so I just slid the heatsink around on the chip surface hoping the old stuff would be good enough (was enough to be sticky).
Anyway... this wasn't the CPU. I don't know what it was. Any printing on it was covered up by the paste. It's almost in the middle of the board between the CPU (which has a large heatsink and fan attached) and the power supply connector.
I had to disconnect the PSU from the board while unplugged from the wall and reconnect it after plugging the power cord into the wall, otherwise not doing so left the condition of the computer the same as before. Once the power was going in and the mainboard connector (actually two, small and large ones) for the power supply was reconnected I got a green solid light from the back of the PSU and all was okay.
Unfortunately, upon moving the computer back to its place on the desk it happened again, blinking green light and no power up from the front button. So apparently if the power ever goes out this will keep happening, but for now the fix (disconnect/reconnect PSU mainboard wires) is good.
I should add that this computer was always on, never off, because it bothered my parents to need to wake the computer up. Probably very lucky it is okay after not having a heatsink on one of the chips for however long a time it was. Or maybe it just happened recently and that caused a nearly immediate shutdown(?).
Hope this helps anyone with one of these HP's if something goes wrong.

Might be a good idea to look inside them anyhow... but then who is going to know to do that without reading things like this message thread here??