I am assuming (please correct me if I am wrong) that you hear the circulation fans spin up, maybe the Optical drive power light blinks on start as normal, but the computer is unresponsive and monitor is dark. In addition, I assume that if you have a laser mouse, the laser lamp on the bottom is dark and does not glow.
It this is correct, it appears that the signal from the power supply is not even reaching the BIOS to start the boot procedure. There is no keyboard recognition beep so that supports the diagnosis. The HP logo is coming from a chip in the monitor, not the BIOS, so the monitor seems to be working. You have reset the BIOS by removing the battery for a few minutes and reinserting it, correct?
What you have is a low-level hardware failure. There are only 4 things that can cause that: a corrupted/failed BIOS chip, a failed power supply, a failed CPU or a failed motherboard. You have attempted a reset of the BIOS, so a corrupted BIOS is ruled out. Unfortunately, this unit is one of these "all-in-ones" that HP manufactured for a while, so testing the power supply on one of these is out of my experience. Perhaps someone else with experience with these can post in.
So the PS is an unknown. If you can access it, and if you have a power supply tester, pull the 24-pin plug off the mainboard and plug it into the tester, then plug in the power cable to the computer.
That leaves the CPU and the mainboard. In all my years of working on computers I have only had two CPUs that failed, but lots of mainboards. Either way, it will probably cost 33% (or more) of a new unit to replace the mainboard or CPU, and that depends of if the mainboards are even available any more.
Perhaps taking it to a dependable local shop might be the best thing at this point. I strongly do not recommend any retail, "box" store. They are notorious for inexperienced help, and eager to sell you their products more than really helping you.
[EDIT]: On thinking about it, one last thing you can try. There must be a way to access the memory modules (aka: "sticks") on that unit, probably from the rear. Possibly a small cover that snaps off or some such. Try removing the stick(s), gently wipe the contacts with a paper towel, and put them back in. I don't think that will be the problem, but at least it's worth a try and will only take a couple of minutes.
Edited by ranchhand_, 26 October 2016 - 08:24 AM.