Your battery is not the problem. I tested my Toshiba Tecra M3 laptop and with the power adapter plugged in it will boot up with a flat battery or even with no battery at all.
Because your laptop showed signs of life only with the new power brick, the old one is probably bad. Rather than throwing out the old power brick, you should troubleshoot it and see if you can repair it.
When you power on your laptop, listen closely. You should hear the hard drive spin up almost immediately. If your laptop has been without power for a long time, it will beep and prompt you to press F1 to reset the date and time.
If your laptop is completely unresponsive, disassemble and check the power button. While it is disassembled, you can inspect for signs of the spill you suspect that your kids committed.
You can find a link to the Tecra M3 manual here:
On page 209 it says this:
The computer will not start.
Make sure you attached the AC adapter and power cord/cable properly or installed a charged battery.
Press and hold the power switch for at least 10 seconds.
If you are using the AC adapter, check that the wall outlet is working by plugging in another device, such as a lamp.
Verify that the computer is on by looking at the on/off indicator. If the indicator is glowing, the computer is on.
If you are using an AC adapter, verify that the computer is receiving power from the external power source by looking at the AC power light. If the indicator is glowing, the computer is connected to a live external power source.