My mom has a Thinkpad 13 2nd Gen whose keyboard just stopped working. I unplugged and replugged the battery in and it worked for a few months and now that won't even help. Its a non-backlit keyboard, so my question it can I replace it with a keyboard for the model that is backlit? She just started school again and I think it would help her to have one that's backlit.
So you know, normally a battery has nothing at all to do with a keyboard working or not. It is likely something was loose or inside the laptop and the issue was resolved when you were messing around inside with the battery.
If this is the case here, it is possible you don't need to replace the keyboard and just need to find the connection problem. With luck that will fix it. If not, then in answer to your changing it question, No, it would not work.
Backlit keyboards require the system have actual power set to go to them directly. You would have to either look into a totally different laptop, that has a backlit keyboard, or consider an external keyboard with backlighting.