Thank you for responding.
Let me out line some of the reasons why you're not getting answers from other members. The first thing is you have not described accurately all the equipment and resources you have available to you.
First of all when you know the exact model number of the computer you wish to power on without reaching down and hitting the power switch button.
Next, it would be helpful to know what is the other computer that the works the way you expected to. Again, the exact make and model are needed.
Why is this? Because the manufacturers don't always stick to the accepted standards in the industry and even the same manufacturer may make variations on established designs used by that manufacturer. This drives everybody crazy trying to figure out why some hardware does not have certain special features.
There are a set of established protocols for power control. One of the specific features has to do with power interruption. If the power from the main circuit of your house or office becomes interrupted, the PC goes dead and this might result in some damage. Or maybe not. In either case, there is a protocol for what to do when the power comes back on. Not all desktop computers have this feature. With a laptop it would not be a matter of concern, because the laptop will automatically switch over to battery power and run on the battery until it becomes discharged.
With a desktop PC the only internal battery runs the clock and it does not have enough power to run anything other than just the system clock. So when you turn it back on it will have the correct time and date.
Now if you're hardware does have complete set up for this kind of thing, it provides another way to turn the computer on after it has been turned off. There is one setting where computer will turn back on if the power is restored. However, the default setting usually is to leave the computer turn all even when the power is restored again.
The difficulty here is the lack of standards among the computer manufacturers and even the use of standards even among computers of the same brand. So that is why we need also specific information about your hardware in order to tell you what's going wrong.
As I mentioned earlier, most desktop PCs, maybe I should say all desktop PCs, have some way of supplying a small amount of power to the reset button. The reset button is a momentary button, it does not latch into on or off position. It is like a doorbell. It just gives a momentary impulse and that is enough to start the computer up. But that only works if there is a standby that provides standby power for the reset switch. I to the said power switch. Wait, I should said power button. One thing that is confusing is picking out the right name is for the different components. I think the proper name is momentary power button. It is not really a switch that goes into a nonphysician or to an authorization and holds the in one of those positions. That kind of pushbutton is called unlatching button and has some kind of mechanical way of holding the button in one position or another. It is kind of like the mechanism used in a ballpoint pen to extend or read track the point of the pen.
Using the search YouTube and find how to replace or repair the power button. Using that information you could create an extension cable for the power button and put it where you want. This is a hardware solution that, I think, works on any modern desktop computer. I believe the power control is now always a momentary pushbutton that connects to the motherboard which connects to a special standby connection from the power supply. Anyway, you can extend that connection and put another power button connected to the same terminals on the motherboard. For just and will the power button and extend the wires over the long enough to where you can reach them. The voltage level on that power button is very low and does not present any kind of hazard. However, to make technical, it violates some of the standards for desktop computers. You're not supposed to have unshielded wires coming out of the computer. Not that it was liable for the computer, but the computer will cause interference to standard radio stations.
Anyway, that is the easy and sure fire hardware method of doing what you want to do. Just extend the power button outside of the case and find a location on your desktop for you can put the power button.
Yes, in theory your supposed to be able to do it with the right kind of software set up and the right kind of hardware set up. The problem is that not all computers have the right kind of hardware for doing power on from the keyboard or the mouse. You can do power all, but you can't do power on if there is no standby power. What I'm trying to say is some desktop PCs do not provide standby power to the keyboard or to the USB connector. So you have to know something about the design of the motherboard. If it does not supply standby power to the keyboard, this not much you can do. The keyboard communicates with the computer only through software. There is no hardware switch on the keyboard that directly ties into the power supply.
I hope this try to clarify what the problem here.
EDIT: Here it is on You Tube.
https://www.youtube.com/watch?v=pOTC9zEcATI