While I would not have expected it to be so black-and-white, let's face it: that battery's REALLY OLD. It has probably been quietly degrading for a long, long time now. Batteries do this.
Obviously you should check connections, try a different charging cable, and download
Coconut Battery (<--link to right version for your machine) to see if I'm right, but I think I'm pretty safe in saying it's time to replace that battery at the very least.
PS. For about 3-4 times the cost of replacing the battery, you could get a much more recent and modern used Intel Mac mini that would bring you much more up to date, so you might want to consider that.