En-el 14

T bought a new D5100 about 4 years ago and the battery(original) does not recharge. I cannot figure out whether the charger is faulty or the battery. Is it possible for the battery to become bad so soon.

Thank you for help with any suggestion you can give me. If I do get a new battery, should I get the EN-El 14 or will the 14A work.

4 years is a long time for a battery, rechargeable batteries only have a certain life and recharge times before they lose their power.

I would put an educated guess on the battery, buy a new one and try, having too many batteries is not a bad thing!!

First, welcome to the forum. 4 years is a long time for a battery and it would not hurt to have a backup anyway,
When you are looking you might want to look at the Wasabi batteries. I have a couple of them for my D7100 / D750 and I can't tell the difference between them and the Nikon batteries other than cost.