Ok the next step would be to isolate the monitor, so see if this problem lies within the monitor. For this, you'll need another video source like another computer or laptop or something you can plug the display into. You seem to have already sort of done this, if I understand correctly, because you have the same issue/behavior whether you connect to the discrete video card or the integrated video card?
I'm starting to suspect the display. You mention that you can try powering on the display and it takes a while to come back. So, maybe it's the backlight. There is also a known issue with some displays and the backlight capacitors going bad. This results in difficult issues getting the backlight to come on and stay on.
So if you don't have another computer or video source to try, you might also be able to look closely to see if the issue is specifically the backlight, where the graphics look fine, but the backlight just is having trouble.
If you think it's the backlight, you can unscrew the monitor housing, and visually inspect the capacitors. You'll be able to see the ones that are swollen or burst, like this:
http://www.lcdparts.net/cap.aspx
I've replaced capacitors on an old Samsung LCD I bought for $10, and it works better than new (the replacement capacitors can be like $3 for a set, and they are higher performance/capacity so your backlight is more responsive).
Here is a nice walk-through:
http://510x.se/notes/posts/Repair_LCD_monitor_with_bad_capacitors/
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