Axial capacitor terminal leads snapped!
The lead was welded to the bottom of the aluminum can.

There's no easy good way to connect to that can without either a spot welder or perhaps conductive epoxy, neither of which would be all that great. 

Ordinary solder won't work at all, and getting it hot enough to solder with special aluminum solder/flux would damage the capacitor/ 

I suggest trying to fit a radial cap in the space, if you can't find another axial one, they're quite cheap. That part also looks very old and electrolytic capacitors do not age well, they're often the first parts to "wear out".