You could try opening your iMac so you can turn it on and see if the diagnostic leds light up.
http://support.apple.com/kb/HT2831
in particular:
Locate the large white arrow in the middle of the computer.
Above this arrow, you'll see four LEDs:
LED 1 indicates that trickle voltage from the power supply has been detected. This LED will be ON when the computer is turned off and your power supply is working correctly.
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UPDATE:
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If the 1st light is on that indicates that the power is flowing to the computer. So it could very well be a logic board failure. At this point i'm not sure trying to repair a logic board failure is worth it.
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Sorry.
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Make sure you pull the HD though if you want to recover your data.
You could try opening your iMac so you can turn it on and see if the diagnostic leds light up.
http://support.apple.com/kb/HT2831
in particular:
Locate the large white arrow in the middle of the computer.
Above this arrow, you'll see four LEDs:
LED 1 indicates that trickle voltage from the power supply has been detected. This LED will be ON when the computer is turned off and your power supply is working correctly.