Ice dispenser flapper door won't close - Whirlpool Side By Side
If its the mechanical style dispenser then it probably is a broken mounting tab. The problem is the mounting tab is part of the complete door. The kit you mentioned was designed to fix it by adding a metal bracket in place of the plastic tab on the door.
If it is a newer electronic style dispenser then the little chute door motor may have failed, but most of the failures I've seen so far with that motor was failure to open, not close.