Laptop won't turn on. Is it my PSU?

First, to any laptop owners, a simple question:
Should the LED on my PSU stay on when it's plugged in an outlet, or just when I turn my laptop on?

I'm praying it's only the PSU who's gone to shit.

More details:
- PC turns on for 2 seconds, starts taking off and then shuts down, nothing on screen.
- There was a power outage earlier today. I'm assuming the battery is now empty, since I was sleeping and I don't turn my computer off. If that's the case, the PSU being busted won't allow me to recharge, but there's still that 1% charge giving it the initial 2-second boot.
- If I unplug the battery, then it won't even turn on for those 2 seconds.

What does /wsr/ say?
It's night here, but I will see if I can find a way to measure the PSU's power tomorrow. I haven't backed anything up in maybe 2 years, I hope everything will be fine.