What to do about a laptop whose screen won't come on.

I have an Alienware laptop, several years old, and the screen won’t come on. I bought a dvi-hdmi cable to connect it to the tv as an external monitor but that didn’t work…probably have to actually tell the computer to use the new hardware I’ve connected? Which I’m not even sure is possible without being able to navigate the screen.

Background:
I got this as a hand-me-down computer. For a long time its first owner could close it without shutting down or putting it in sleep mode and it would be fine. Then it developed a habit of not turning its screen back on when closed and opened again. I would have to try to power it down and up a few times and it would finally come on. I stopped closing it unless it was completely shut off, but eventually it did get closed without powering down, and now no manner of opening, restarting, swearing, wiggling of the hinges or smashing of keys will make the screen turn on. It powers up and runs, but the screen is completely black. No juice. I did just notice that it doesn’t run for long this way before turning off on its own.

Is there anything I can do to connect a separate monitor or do I just need to have my data retrieved and call it a loss?