Suggest perform the simple test to see if the problem is the Power On switch, or the Power Supply.
This requires bypassing the Power On switch.
DOES NOT involve cutting any wires, or splicing any wires.
Before we go on I wish you to know this;
The metal case of the Power Supply, retains the dangerous high voltages inside.
100 to 240 Volts AC, depending on what country you are in.
The Power Supply is a Converter.
It converts the high AC voltages down into 3 low DC Voltages;
3.3 Volts, 5 Volts, and 12 Volts.
In comparison two D cell flashlight batteries produce 3 Volts DC.
The power cables coming out of the Power Supply, carry either
3.3 Volts, or 5 Volts, or 12 Volts; DC.
(Orange = 3.3 Volts. Red = 5 Volts. Yellow = 12 Volts)
The Power On switch deals with the Soft Power On circuit.
This circuit uses 5 Volts DC.
Posted, because I wanted you to know, I AM thinking of your safety at all times.
However like all connections made when dealing with electricity, there may be a small Spark.
You may wish to wear a glove on the hand, that deals with the jumper wire.
Let's look at the motherboard used in a Compaq Presario CQ5110F Desktop PC,
HP Support > Compaq Presario CQ5110F Desktop PC > Main Support page,http://h10025.www1.hp.com/ewfrf/wc/product?cc=us&lc=en&dlc=en&product=3978596
Click on Product information > Product specifications > Motherboard Specifications, M2N68-LA (Narra5),
Motherboard is shown with no hardware components attached, nor cables. I'm just using it for Clarification. Everything is supposed to be attached when testing.
Look all the way to the Right side, and to the long black rectangular connector. Now look at the long whitish/yellowish connector under it.
Has two columns of 12 Socket Holes.
This is where your 24-pin ATX main power cable plugs in.
This is a general example of a 24-pin ATX main power cable, and it's respective connector on the motherboard,http://www.playtool.com/pages/psuconnectors/connectors.html#atxmain24
[Note color of connectors does NOT matter ]
You need the main power cable plugged in, as shown in the photo to the far Right.
Observe the Green wire. This is the Soft Power On wire.
-> ANY Black wire you see is a Ground wire.
The preferred jumper wire is a paperclip.
Straighten it out, then bend into a U-shape. Turn the U shape over in your mind.
The Top of the U is wrapped a few times, with black plastic electrical tape.
This taped area is for your fingers, and thumb to hold onto.
The 'legs' are what is used for a jumper.
One leg goes down into the socket hole, with the Green wire.
The BACK of the connector is where the wires go in.
The leg goes down in the Back.
It slides down in the socket, RIGHT NEXT TO the insulation of the Green wire; and touches a metal female terminal.
This is an example of the female metal terminal. Just posting for edification,http://www.mcmelectronics.com/product/28-9420&utm_medium=Affiliate&ref=cj&utm_campaign=CommissionJunction&utm_source=CommissionJunction
At the end of every wire going in the main power cable's connector, is one of these female metal terminals.
You can barely make out the round openings, in the middle photo of the Playtool link above.
For this reason you need to Ensure that the leg of the jumper wire, is indeed touching that metal female terminal.
The other leg of the jumper wire goes down into ANY Black wire socket hole.
Power Supply plugged into power.
The contact time is no more than 2 seconds. This is because the Power On switch is a Momentary Contact Switch.
IF, the Power Supply turns on, you have a bad Power On switch.
IF, the Power Supply does NOT turn on, you have a bad Power Supply.
This baby works in MOST personal computers,http://www.directron.com/atxswitch.html
The Power On switch is located INSIDE the plastic assembly, of the Power On Button.
I use a hair dryer set to low heat, and constantly, slowly, move the hair dryer nozzle; back and forth on the plastic of the Power On Button assembly.
Helps to soften the plastic enough, so that the assembly will spread far enough, to get the Power On switch OUT.
Awaiting ANY additional questions. Please post in a Comment.
(Remember, there is no such thing as a stupid question, just one that hasn't been asked yet)
Can guide in replacing the Power On switch, (Remove Front Panel), and in replacing the Power Supply.
[When you are finished working on a desktop computer, remove and reinstall the Ram Memory modules. ('Stick')
They always seem to get bumped loose.
Problem is you may not see it. In fact betting you won't.
Best method to ensure they are seated tightly, and correctly; is to remove and reinstall.
If you move the cables with the computer, you can lay it on it's side, opening of computer case -> UP; for more ease of working on it.
Your computer can lay on it's side, or upside down; and will work ]