Hello there,
If the backlight is out, the screen image will still show up but there is no light to shine through it so you can normally see it.
The test that I've used is to get into a bright area(outdoors on a sunny day) and angle the screen around to perhaps see any of the displayed images.... or if indoors, get a flashlight and shine it into the screen at an angle and get your face up close and move the light around the screen rather close.
This may look and feel rather awkward, but I'll say that it has served its purpose many times in the past for myself in such diagnostics.

Should it be the LCD screen that has failed, but the backlight is still working, in a darker room you'd notice the screen go from completely black to a slightly lit-up "dark gray"... which would be a noticeable change, at least, in a darker room.

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As for the external monitor that you've got... I'd look into the blue 'Function-Key combination' symbols up alongside the F1-F12 keys... various laptops have a "Fn + F(1-12)" combination that will enable and output to an external monitor. Look for a blue-ish symbol that looks like two screens side by side, with a vertical bar in between them.
Alternatively, I've known laptops to just enable a secondary screen and run with it, if you plug the screen into the laptop before you turn the laptop on. Once you turn the laptop on it will duplicate exactly what the laptop is(or should be) showing.
However, i have seen Windows disable an external monitor once it gets to the login screen if it's not used to it being there - this is just to warn you that it may happen like that.
... Sheesh, also... some laptop bios's have settings that will enable or disable the use of an external monitor by default... that would be "the pits" in this case...

I hope that works out for you... follow up with any results and we'll try to go from there.