In addition to @Brad's Penetrene tip, try using a rubber band for grip.
I shared the same tip on this discussion - https://www.workshop.com.au/t5/Interiors/What-s-the-best-D-I-Y-advice-you-ve-ever-received/td-p/222 - which has lots of good advice.
Good luck with the project.
@Focusmarine I'm taking a different interpretation of your problem than the other Workshoppers, so feel free to ignore it if I'm off target.
I originally thought of drilling many small holes, as close as possible around the broken shaft, but then later lucked upon this YouTube vid:
Great vid, thanks for sharing. Just to let you know I've edited your post so the video is embedded rather than just a link. When you have a YouTube video you'd like to embed in a post, you just need to click on the "Insert a video" button above the text editor. It's the icon that looks like a film strip.
@Jason thanks mate for the tip on imbedding clips,you're a true champion.
I think I'll create a document file on my 'puter, so that I can save all of your helpful tips, that aren't listed on the site help page. * inserts light bulb emoticon
@Focusmarine I'm delighted that the vid could be helpful, you've made me very happy.