Why are you trying to switch it off from the power supply and not the button on the front of the computer?
The wires coming from the front panel from the power switch is only 2 wires. Usually blue and white I believe. These connected to your motherboard is what allows the computer to be turned on because it sends a signal the the PSU to turn on basically. Does the front panel button not work or is it simply not present?
Also if that button doesn't work have you tried holding it down for 5 seconds to see if it powers off? Have you gone into the BIOS and tried changing the properties for the power button because you can make it so the power button does something else besides turn the computer on or off, so it could even possibly be disabled so that the normal power button couldn't be used.
Considering if all above info is non-applicable. It sounds like you have a "test-psu' so to speak. I've made one before and basically you use it to test fans and such without having to plug your PSU into the motherboard and have the motherboard send the signal to the PSU to give power. To make a PSU into a test PSU a couple wires are cut from the normal 20pin connector that goes to your motherboard. A green and black wire. Those two wires are then connected together thus creating a PSU that will power on when the switch is on.
So it sounds like essentially it was turned into a test PSU and they took it even further to unwire the switch from the back of the PSU resulting in a PSU that basically is on forever.
Here are the directions on how a test psu is made so you can perhaps get an idea of what may have happened to your PSU.
Look to see if wires have been cut from the 20pin connector, the green and black one mentioned in the article. If they are, then theres your reason for the PSU coming on as soon as it's plugged in. Then you'll need to look at your PSU for the switch wires and I guess reattach them accordingly. The Test PSU mod does NOT have any effect on the switch and the basically the switch is the only thing that allows you to turn it on and off so now in your instance, what's going on with the switch remains a mystery.
Someone must have modded it so that it just remains on at all times. Check that test psu article to get a little idea of possible wiring and recheck wiring for the switch.
EDIT: I just noticed something else
Ihave several,but there are three bundles of cables that attach to this mobo,unlike my others,which seem to have only two.
What bundles are you talking about? For a normal AMD Based machine there's the 20pin connector that's 2 rows of 10 with oddly shaped holes, and a square 4 pin connector that also connects to the mobo that powers just the processor. On most newer PSU's it also has a 3rd wire, but I believe this is for Intel Based machines, but they put it there so you can use the power supply for either Intel or AMD. The 4pin connector I'm talking about doesn't look like your normal 4pin molex connector which is 4 pins in a straight line. The one I'm talking about is in a square shape. 2 connectors ontop of 2 to form a perfect square. The Remaining bundle if I'm assuming we're thinking of the same one is not needed and you shouldn't really worry about it, especially if your board doesn't need it. I'm pretty sure there's no motherboard that requires 3 power connections to it..only 2...there are only 3 connections so that it is compatible with both Intel and AMD based machines