When you unplug the power cable, be sure that nothing else is plugged in. No external drive, ethernet, USB cables, unplug everything. 5 or ten seconds will be enough, but let it set for a couple of minutes with no connections.
Press and release the power button (still nothing connected)
Wait another 10 seconds, then plug in power cord ONLY. Wait another 10 seconds.
Now, press and release the power button.
You should hear sounds, like the boot chime, or listen carefully for the hard drive to spin up. You may even hear the fan spin.
Connect a USB wired keyboard, press the caps lock key. Does that key light up?
If you get no response,yYou will need to open the iMac so you can see the internal test LEDs.
That will tell you very quickly if you have a power supply failure, or some other issue, any of which will need some hardware repairs.
If you are uncomfortable with opening up your iMac, take it to a Apple-authorized service shop.