If you can get that laptop to project a screen to an attached monitor using the HDMI or VGA cable that would prove that it's just the LCD panel that's gone.
When you connected the HDMI to the other screen what else did you do? or did you expect it to just come on as soon as you plugged it in?
Have you connected the laptop thru the HDMI or VGA connector to another Monitor before this?
Yeah, you need to cycle the Fn and F5 key a few times. Actually 2 times, but you can continue doing if till you feel that you've exhausted the options.
It might be repairable, you can try first by replacing the LCD panel, which is not too expensive, most probably cheaper than the GPU. And then if that works up.... great for you. If not, you can go onto one of those trading site for a salvaged 8800M GTS or something which will be compatible with the socket of that mobo and the power and the cooling option sufficient but higher than the 8800M GTS and give it a try.
But because you get the Windows Startup Tune, I doubt if the GPU's gone bad, it's most likely the LCD only.