Here is an example of how we approach a repair.  In this case the homeowner had attempted a repair, and in the process made quite a mess and frankly made our repair more difficult.  We removed the tile, removed the underlayment, replaced the damaged wood, installed a high quality base sheet, installed new flashing and counter flashing, installed heat applied modified bitumen cap sheet, reinstalled the tile and provided more tile for those tile broken, and painted the tile to match.  We don't do shortcuts.