If one broke the rest are next. Fix this one and do the same thing to all of them before they do.
Drill pilot holes (red dots) and attach with short screws (be careful not to punch though the side of the cabinet).
Since this one is broken it needs some screws where the blue dots are too. On the rest of them, I'd do screws in the two red places and one blue one about an inch or two from the original screw.
But other than on the broken one, a single screw into the 'cup' should do (red dot on the left is inside the 'cup'). Use the largest size pan or round head that will fit in the 'cup' and that slot in the plastic (like the original ones).
Drill the plastic piece first and then use it to template where the pilot holes go. You only get one shot before it becomes much more difficult to realign it. Doing the rest of them before they break will be much easier.

I really can't believe they designed it this way, without a screw in the cap, leaving it with a 2" fulcrum.
You can find all the parts you need at any store that sells these or their equivalent: (just make sure they fit inside)
#8 X 1/2 in. RWH Sharp Point K-Lath Screw (source)
