if the screen is going blank then it's probably the backlight connector, most likely where it plugs onto the inverter. (have a look at it under bright light or shine a torch at the screen and you can probably still just see what on it)
The inverter is normally located just above the keyboard on most laptops and looks like this
http://cobolhacker.com/images/content/inverter _and_pencil.jpg
you will have to strip the laptop down a bit to gain access to it but as you haven't said what lappy it is I can't really offer advice on how to do that for you. but if you google "[your laptop model] change inverter" you will probably find instructions to get to it.
make sure you have the power disconnected and the battery removed when going anywhere near the inverter, those things are putting out about 700volts and will give you a nasty shock (it hurts... trust me!)