Yeah well this is definietly a problem.
The fact that it doesn't even turn on when the power is inserted suggests to me its something within the laptop itself as when power is going through from the mains, its should work as of normal without using the battery.
Though when you turn it on without the power pack, the computer might be right in thinking there is no power in the battery, but because there is something wrong with the connection to power i the laptop itself, its not getting access to any to work
So i'd get the laptop checked out, though I've heard on a few laptops that the battery has to be working to power the laptop even when in mains mode though with yours it might not be the case. I wouldn't advise you as of yet to buy a new battery (though it'd work for the suggested 2 hours or so) I couldn't say that if you plugged up the power, it would charge.
So get it checked out if you can, the insides, and hopefully it iwll be this so you can just buy a new battery and make sure it never runs completly flat, though its more likley to be the power terminal in it.
I hope this helps