1) Computer unplugged from power.
2) Observe Anti-Static Precautions
Your body carries Static electricity. Static WILL fry out (Short Circuit) the hardware components inside a computer.
Computer on a table, computer unplugged from power, computer case open.
TOUCH the metal frame of the open computer case to relieve your body of Static, Before reaching into the computer.
Should you leave your computer in the middle of working on it, upon your return be SURE to Touch the metal frame again.
At the back of the computer case, perpendicular to the Support Bracket, is a Card Retention Bracket.
The small metal plate that holds add-in cards in.
(Ethernet Card, Modem Card, Graphics Card, etc)
Lift the card release lever up to release the Card Retention Bracket.
Swing the bracket up.
Now swing the end of the Support Bracket up that is on the Card Retention Bracket side.
The other end of the Support Bracket is on two pins. With the Card Retention Bracket in an upright position it will lift off of those two pins.
(The end of the Card Retention Bracket that is on those two pins, has half moon shapes that go over the pins. It may be that you have to bring the bracket more down towards the computer, in order to remove it from the computer)
Now on to the real problem.
Are you going to bypass the Power On switch to see if the problem is the switch, or the Power Supply?
This has nothing to do with the Power On switch, or it's wires.
A jumper wire is used in the 24-pin ATX main power cable's connector.
It is used to jump from the Green (Soft Power On) wire to ANY Black wire. (ANY Black wire is a Ground wire)
The 24-pin ATX main power cable remains plugged into the motherboard. The computer (Power Supply) is plugged into power.
The jumper wire goes down into the Back of the 24-pin ATX main power cable's connector.http://support.dell.com/support/edocs/systems/insp531s/en/OM/html/parts.htm#wp1483829
P1 in the illustration denotes the 24-pin ATX main power cable, coming from the Power Supply.
This is a more in-depth look at a typical 24-pin ATX main power cable, it's respective connector, and the matching connector on the motherboard,http://www.playtool.com/pages/psuconnectors/connectors.html#atxmain24
The view to the far Right shows the 24-pin ATX main power cable plugged into the motherboard, and shows the Green wire.
(Green wire = Soft Power On wire. Also abbreviated as PS_ON)
ANY Black wire you see is a Ground wire.
The jumper wire is a paper clip straightened out, then bent into a U-shape.
Black electrical tape is wrapped a few times around the middle of the U, for your fingers to hang onto.
One 'leg' of the jumper wire is inserted down into the socket hole with the Green wire, RIGHT NEXT to the wire.
The other leg is inserted down into ANY socket hole that has a Black wire.
The jumper wire Must go down into the socket hole far enough to touch a metal connector.
At the end of every wire going into the ATX main power cable's connector, is a Molex female crimped terminal connector,http://www.alibaba.com/product-gs/336134982/molex_female_crimp_terminal_connector/showimage.html
(The depth is approximately half the length of a fingernail)
The contact period is around 2 seconds.
There may be a spark.
The voltage is 5 Volts DC
Two C, or D cell flashlight batteries produce 3 Volts DC.
IF, the Power Supply comes on, you have a bad Power On switch.
IF, the Power Supply does Not come on you have a bad Power Supply.
This is an example of a Power Supply for a Dell Inspiron 531s desktop computer,http://www.censuspc.com/product-pr-11141.html
For questions post in a Comment.