How to retrieve broken off end of gear cable from inside Shimano 105, 11 gear, lever?

If I understand correctly, you have access to the end of the cable coming out of the shifter, but not the end inside the shifter, with the metal cylinder. To get access to the cylinder end, you need to pull on the cable end with pliers and then click through the gears until it is in what would be the smallest cog. At this point, you should be able to peel back the hood to grab the cylinder (maybe with needle nose pliers or a sharp spoke) to pull it through.

I've done this with a RSX shifter. It was a full on dismantle and rebuild to get the mushroom out, because it had wedged and the wire stubs splayed out stopping it from shaking loose.

The real solution is to periodically change your gear inner cables when using brifters. If shifting starts getting really tenuous, that's your warning to change the inner.

Do check out Disassembling Shimano integrated gear shifters (brifters) and STI shifter sometimes doesn't catch in cooler weather

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 "teeth" point backwards in the assembled brifter.