Satellite A300 PSAG8E does not charge

Question

I have a Toshiba Satalite A300 PSAG8E which does not charge.
I was handed it a few days ago by someone claiming that it doesn't work or turn on -- vague details.
Upon receipt it turned on, asked whether to start windows normally or not, then booted absolutely fine, it powered from the mains charger, no problem.
I left it on for a while doing various tasks.
I came to it a few days later to find it had turned itself off.
I powered up again, at which point it asked whether to start windows normally or not. I started normally and it came on fine.
Since then I have been unable to power it from the mains.
Why won't it charge now?

Answer 1

Thanks for your responses so far. Much appreciated. Not to be rude guys, but I thought I was clear enough. However, I can understand the confusion as the situation is unusual.
The problem is simple:
When the laptop is plugged into the mains, the battery does not charge. The laptop does not power on. The situation is this: It only does this sometimes.
What I have tried:
Wiggling the cable, in all sorts of ways to see if there's a possible connection issue. This does not appear to be the issue. It seems to be simply on or off.
I tried resetting BIOS to default before the battery went dead last time. Still no power.
http://aps2.toshiba-tro.de/kb0/FAQ6201DK0004R01.htm
# Make sure the battery has a full charge.
- No, it has run flat since it charged last. Unable to recharge.
# Try plugging the AC adapter into the computer.
- Tried. Nothing.
# Try plugging the AC adapter into a different outlet.
- Tried. Nothing.
# Try pressing the power switch for ten seconds.
- Tried. Nothing.
# Make sure the power switch is not physically locked off, like a cover protecting it.
- Checked.
# Try removing the battery and the AC adapter, then plug only the AC adapter in and try powering on.
- Tried. Nothing.
# Make sure you have the proper AC adapter, and only use the one that shipped with the computer.
- Yes.
# Is the computer turning on, but the display is blank or being routed to a monitor that you can't see? If so, try using the FN+F5 keys.
- No. It is not and does not turn on.
# If the computer is on, but no display or won't power off, you can press and hold the power button for ten seconds (until it powers off), then try powering on again.
- No. It is not and does not turn on.
# Make sure you remove all extra hard ware, especially extra memory even if it shipped that way from the dealer. The memory could be loose, or could have gone bad.
- Checked. Nothing.
# Look at the lights on the computer for clues. Are you getting AC power? Is a light blinking? If there is a blinking light it can be an indication that Power supply Micro controller detects an error. In this case please contact ASP (Authorized service partner) in your country.
- No lights. When battery is in, battery light blinks when you press the power button because the battery is dead.
What I will try next: Detach the battery and power cord overnight.
If I manage to get it to power up, update BIOS.
Any further advice?