2002 IBM ThinkPad T23 battery problems/won't run without power
It doesn't look to me that you're doing anything wrong there.
Was there a specific installation requirement for the battery?
I recently changed my battery on my IBM/Lenovo and all that was necessary, was to let it charge fully before usage.
If you say your battery was charged for "a few days", then I assume there is something wrong with the battery. Next to your taskbar you should see a battery symbol. Double-click on it and you should see the Thinkpad Power Manager. If there is a tab called "Battery Information", see if that gives you more details. Look at the Manufacturing date!
If you really can't run the laptop without being connected to a power source, I would definitely return the battery or at least call the vendor. If your computer is running fine otherwise, there should be nothing wrong with the machine itself.
If you can take a screenshot of that battery information window adn post it here, I am happy to look at it.
You're doing the right things when you try to use the laptop without an external power source - that's the whole idea, isn't it?