That port will plug into a standard monitor (or projector, as you know).
From your description the backlight in your laptop's screen is not functioning. This is commonly one of two things: The light itself or the inverter board that powers it.
I've replaced the inverter in a couple of laptops & LCD desktop screens. Pretty easy & usually quite cheap. If you're comfortable disasembling your laptop then you can usually replace the inverter board quite easily.
If you're *really* lucky it might just be a cable in to or out of the board that's come loose and resocketing its plugs will fix it. Don't fiddle with the inverter with the power on - it's a high voltage device.