Which year/model is it exactly?
I see you said you tried plugging it in while holding down the power.
Try unplugging everything (Power cord, USB cables) wait about 15 seconds, plug the power cable back in and then try pressing the power button.
Listen closely, any sign of a fan/HDD running? If that model has a LED on the front, does it light up at all?
Then again you did say it shows no sign of power.
If your able to open the unit, some IMacs have Diagnostic LEDS on the board, are you able to access those?