Check tower that will not power's, Power On button, by bypassing it. (I kinda' liked the heading)
IF the Power Supply is good, the Power Supply will turn on. This would mean you have a bad Power On switch.
(Happens. They do wear out. Replacement is about $5 for the switch) http://www.directron.com/atxswitch.html
HOWEVER, (Not shouting) there is another switch also. It's a Smart Cover Lock. If this switch is bad the Power On switch won't work.
If you're not familiar with computer hardware, it may seem like I'm typing in a foreign language here.
I understand this. Let me try to clarify.
When you press the Power on switch, you are making a Momentary Contact, that momentarily closes a circuit.
You can see the Power On switch is a momentary contact switch, by the fact that it is spring loaded.
The Power Supply when plugged into AC electricity, always keeps a 5 Volt standby power. (5 Volts DC)
Pressing the Power On button, closes this 5 Volt circuit momentarily, and the 5 volts turns the Power Supply on.
This is the Soft Power On.
It was brought in so that the Power On switch could be relocated to the Front Panel of the computer, and not mounted inside the Power Supply.
The older computers that do not have a Soft Power On feature, have a metal rod that one end is pushed by the Power On button, and the other end of the rod pushes against a Power On switch, located inside the Power Supply.
The Smart Cover Lock is a chassis intrusion switch.
If the cover is removed, the power supply cannot be turned on. (Or also said as the computer cannot be turned on)
To readily view where the Smart Cover Lock is located, and a view of the lock itself, I would like you to download the maintenance and service guide.http://h20000.www2.hp.com/bizsupport/TechSupport/DocumentIndex.jsp?contentType=SupportManual&lang=en&cc=us&docIndexId=64180&taskId=101&prodTypeId=12454&prodSeriesId=96266
On this page scroll down to the heading - Service and maintenance information, and under the subheading - Title, left-click on this file name,
Desktop EN Series of Personal Computers Maintenance and Service Guide
This is a PDF file download. It may take a few seconds before you see the first page. It may take longer for the file to download all the way.
Suggest letting the file download all the way, before looking through it.
There are three different model versions of the Compaq Deskpro EN series.
The Small Form Factor, (SFF), model
The Desktop model
The MiniTower model.
You indicated 'Tower', so I'm going to assume yours is a MiniTower model.
Information for the SFF, and Desktop models, is different as to where the Smart Cover Lock is located.
The Smart Cover Lock is a solenoid switch. When the plunger part of the solenoid is out, the switch is open. No voltage passes through, hence the computer doesn't come on.
The Smart Cover Lock's plunger rests up against the computer case. When the case is open, the plunger comes out. This makes the switch a chassis intrusion switch.
If it were I, I would remove the Smart Cover Lock out of the system. (I'd cut it out, and use heat shrink tubing, and solder for the wires)
To bypass the Power On switch:
The Compaq EN series of desktop computers use a 20-pin ATX main power cable. Looks like this,http://www.playtool.com/pages/psuconnectors/connectors.html#atxmain20
Note the photo in the middle. The front of the connector is facing you. The back of the connector is where the wires come in.
With the 20-pin ATX main power cable connector, connected to the motherboard as shown in the photo to the right, a jumper wire is used for the Soft Power On wire.
In the photo to the right, the Soft Power On wire is shown more clearly.
The Soft Power On wire is the Green wire.
(PS-ON, or pin number 14 in the chart below)
A jumper wire is connected to the Green wire's terminal pin, and over to ANY Black wire's terminal pin. MOMENTARILY.
ANY Black wire you see is a Ground wire. By using the jumper wire, you are completing the Soft Power On circuit, and bypassing the Power On switch.
Wires that come into the back of the 20-pin ATX main power cable connector, end in a metal male terminal pin.http://www.mcmelectronics.com/product/WALDOM-ELECTRONICS-02-09-2118-/28-9405
The insulation of the wire is right up against the terminal pin, and the terminal pin is way down in it's respective socket hole of the connector.
The jumper wire is made out of a an insulated wire, that is the gauge of the Soft Power On wire, or larger.
The insulation is stripped back off of each end of the jumper wire, about 1/2 inch.
The exposed wire strands are twisted together.
The bare end of the wire must be able to go down into a socket hole, from the back of the 20-pin ATX main power cable connector, and touch the metal terminal pin way down in the bottom.
You may have to split the strands before twisting, and just use half of the strands, or less, so it will go down in the socket hole. There will be a wire in the socket hole, so it needs to be able to squeeze past it.
The jumper wire is bent into a U-shape. You hold the wire by the insulation.
My method is to have the computer unplugged from power.
I test the wire leads to make sure they will go down in the socket hole far enough, to touch the metal terminal pin.
You need to be sure.
I then leave one end stuck down in the socket hole for the Soft Power On wire, (Green), and have the other end ready to go down into a Ground wire socket hole.
ANY Black wire is a Ground wire, doesn't matter which Black wire's terminal pin you touch.
Now plug the computer into power, and use the jumper wire.
You are only making a Momentary contact. You are just touching briefly for no more than 2 seconds.
Warning you in advance that there may be a spark.
This is 5 Volts. A common flashlight uses 3 volts.
Have any questions concerning this, or need additional information, please state so in a Comment.