How to retrieve broken off end of gear cable from inside Shimano 105, 11 gear, lever?
I just got done doing this to my 5800-series brifter. Looking at it, it seems to be kind of fortified against amateur disassembly.
However, I discovered that this took a lot less disassembly than Criggie's answer had led me to believe. Indeed, you could do it with the brifter still on the handlebar.
All it took is undoing one screw on the underside of the lever:

This let me remove the two bottom parts of the brifter's case and access the part of the mechanism the stub of wire was in:

As a reassembly note, the larger piece is backwards in the image, those &quot;teeth&quot; point backwards in the assembled brifter.