From the sounds of it I suspect your real issue is the displays backlight.
What I think is confusing you is the external display is not working. Are you sure you have it setup correctly and using the good cable?
Instead of worrying about the external display take a flashlight and holding it on the display at a sharp angle are you able to see the faint image of your desktop icons? If you are they the display is working correctly (which the diagnostic LED's are also telling me as well).
Here is the IFIXIT guide to replace the backlight driver board: iMac Intel 21.5" EMC 2428 LED Driver Board Replacement. Before you go down this path make sure the cable connections are correct (if you opened the system).