I can't tell from the photos: Did the main shaft break inside the sleeve?

I've been using a repaired Silca pump for about a year after the shaft snapped half. I found some thin sheet aluminum at the hardware store, cut it to the length of the circumference of the pump shaft, wrapped it around the cracked section, glued it in place and then wrapped it in electrical tape. (It's a black Silca, so the electrical tape is only obvious from up close. Sounds uglier than it looks. I'll replace it eventually when I find one in the right size and at the right price.) In your case, the sleeve might work just in the same way as the the aluminum sheet I used. If you applied strong glue around the inside of the aluminum sleeve and the edge of the break, it might just stay in place and create a good seal. Worth a try, though, if you are able to fix it, don't be surprised if you feel a slight bump in the pumping action as the rubber plug moves over the repaired area.