Those split shafts are notorious for breaking off. Usually, one of the halves get bent or becomes loose. Someone tries to fix by straightening it, or attempts to spread them to get a tighter fit, often by inserting a flat screw driver tip to wedge them apart. Except they are brittle, like pot metal, and don't like bending so they break off instead.
Here is how I would fix it. Remove the pot and cut off the broken end(s). Find a donor pot (or just buy a new pot with a similar sized knurled shaft, usually 1/4" or 6mm). Confirm it fits your knob. Trim off a similar donor length so that the new piece plus the original pot will become the same length as original. Go to your nearest hobby store, or even a hardware store and buy an appropriately sized brass tube that will slip over the shaft. If the pot is for volume/balance/treble/bass -- you can just use superglue and glue the parts together. If the shaft is for something that requires more turning force, such as on a function selector, then use epoxy such as jb weld.
By the way, the easiest way to cut them is with a Dremel type of cutting tool, even battery operated ones. Will cut them in seconds and additional attachments can be used to grind/square and chamfer or deburr the ends.