How can I repair a broken pivot hinge for a vanity mirror?
I have the same problem. These doors are crap.  The backings are 1/8" fiber board glued to the mirror glass.  This 'fake' wood is warping from bathroom moisture, hence the mirrors are becoming unglued and starting to flop.  I am considering replacing this cabinet instead (with a higher quality more pricey one, sigh!), since the 2nd (of 3) door hinges broke in exactly the same way as described in original post, and the third will fail soon, I'm sure.

Having said that, I have a stopgap repair that I do NOT recommend, since I'm petrified this (or the rest of the door) will come apart at any time.  Caveat emptor!




The fix I made was to salvage hinges from a previous glass doored cabinet. You might have to order your own (see below). The bracket on the hinge that attached to the door has a cross section shaped like a "U", or an "L" with a toe on the mirror side (right) and two set screws on the back (left) to pinch into the door board.  Because the base of the "U" was 1/4" I had to glue another 1/8" strip of strong dense wood to the back of the fiber board.  This was ok as it gives more for the set screws to bite into.  I carefully wedged the toe between the glass and the fiberboard (bad, as this further separates the mirror from the fiber board, ugh! and the metal on glass might chip if bumped).
Note that the new bracket "U" height has to be short enough to clear the old rivets.  I had to file away some of the old hardware to make this all fit. This fix has lasted several years of "careful use"... but I wouldn't use around small children and I'm not gonna fix this again!

Glass Door Pivot Hinges: (woodworker.com)