I can't tell what that broken part is, but if I had to guess, I'd say it's some kind of cable housing ferrule or reducer.




If you replace your cable adjuster with a Mickey, you should be good to go. Whether the inside diameter of the end that accepts the cable housing has room for a ferrule remains to be seen. If one will fit, I'd use it. But it might require the cable to feed into it as pictured below:




Hope this helps!