How to retrieve broken off end of gear cable from inside Shimano 105, 11 gear, lever? The gear cable was badly frayed in the handle system, where it bends through 90 deg. I had to cut it to remove the cable...but in doing so, the tip with the cable end (about 10mm long), has moved on round the corner and on into the lower part of the handle where it is very inaccessible. 
I am able to get hold of the cable end with pliers but unable to pull it back, even with the gear changer set in it's highest gear because the nipple is stopping it.
Any ideas?
Do bike engineers in UK shops take these brifter gear levers apart or do they just advise replacement?
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.