Sorry😫 the connector will need to be replaced! The logic board pads still has the broken connectors pins! These can be removed with a soldering iron or a hot air rework station.
Sounds like the battery was in need of replacement and given the systems age it’s long since in need as battery’s don’t have that long a life just by time besides cycle count.
To hep put this into perspective I too need a new battery for my 2012 MacBook which if I remember will be the sixth battery it has had!