@Madasa...
Please read through this before you actually do anything and post back with any questions if you're unsure about it.
The front cover will actually come off. It's probably got a few "hidden" plastic tabs/tangs accessible from just inside that metal frame and it will either snap loose completely or be hinged on one side.
Once it pops off you'll be able to see clearly which wires go to the power switch. Make note of them.
FWIW, the I/O switch on the back of the Power Supply simply turns it on and energizes its coils. I am pretty sure that is an ATX power supply and it looks to be pretty generic. Hard to tell since there is not enough detail in the pics especially of the connectors and the MB layout.
Once the PS is turned on via that I/O switch, the transformer comes to life so it's ready to go, and just a wee circuit on the MB is powered. When you press the power switch, it simply closes that circuit on the MB (think of it like a relay), which then fires off the rest of the circuits and power is supplied both to the MB as well as the drives, fans, ports, etc.
There are roughly three possibilities. The easiest to check is the front panel switch itself. Once you figure out which pair of wires goes to that switch (pop off that front cover!), trace them to the motherboard. They will connect to two little metal posts and should just have a little connector for them. Turn on the I/O switch, then NOTE CAREFULLY where those two wires attach, then lift up the connector. You should not have to remove it all the way, but if you do, that's the "note carefully" part so you don't miss where it belongs. Once you have exposed the bottom part of those two posts by lifting the connector a little, you can then use a very small piece of metal (tweezers are fine but you can probably also use a straightened paper clip) with the end you hold covered by electrical tape -- this circuit is very low voltage and current, but just in case you touch something you shouldn't -- and short those two posts (and only those two!) together. That's the same thing as what the switch is (or should be) doing.
If you hear lots of "familiar" noise then you know the problem is the front panel switch. Most are just a momentary on/momentary off, so you can find a suitable replacement online. It won't look pretty but you won't care, right?
If, however, shorting the posts does nothing, push the connector back down, as the panel switch should be okay, and I would next try swapping the power supply itself. A used one is fine, and "generally speaking" you just want an old, basic ATX power supply (easy to tell that one is old because it has no SATA connectors, but it does have a floppy connector). Based on the PN in your pic that's only a 250W, so be sure you get at least that. If you have a computer repair store in your area they probably have a bunch of used ones laying around they'll part with for a song. Make a note of all of your connectors and replace them one at a time. You may not even have to remove your existing one if the replacement one has long enough wires. Once it's all connected, then plug your AC power cord into it, if it has an I/O switch turn that on, then press the front power button while crossing your fingers.
If that doesn't do it, then the third possibility (now almost a certainty) is the mainboard. Since you won't be able to find one (if you do it will be someone like you that has it because they want it, so they won't part with it) and if you do it will be costly, then just buy an old XP computer. They are all over ebay. Then plug your hard drive into it and copy your music files (from whatever drive letter that shows up as) to the C drive on the "new" computer, and you're good to go. Copy them to a thumb drive as well since the drives are always a weak link. Note: You won't be able to successfully swap your drive in place of the drive in another XP computer and have it work, so don't bother trying. Even if it boots it will not be reliable. Just use your old drive to copy off your music and other files you want to keep.
Overall, if it's not that front panel switch, you could just jump to getting another XP box instead of trying another Power Supply. That's probably what I would do in your circumstances (though in mine, I'd dig around in my pile of crap until I found another PS).
Good luck.