iPod Mini works when charging. Shuts off when unplugged.

More info required, what generation/size is it?

did the connector break off the logic board or wire break in the connector?

Measure the open circuit voltage on the battery directly with a volt or multimeter ,nominal voltage for most i pod batteries is 3.7 volts

if you find the volts are lower than 3.0v its flat or defective

if its lower than 1.0v consider the battery needs to be replaced

although ifixit currently dont list these batteries ,they are widely available on major auction sites