Whirlpool has a repair kit for that it's a metal bracket that gets put in place or where the tabs are you actually break the other tab off to mount the kit The part number has slipped my mind at the present. Years ago When this first happened whirlpool was replaCING DOORS UNDER WARRANTY buit the is a repair kit available It made a lot of sense when they designed thier S x S's with plastic tabs holding a steel pin lol of cousre it's gonna break under pressure what causes it to break is the flapper door getting frozen to the chute opening from low useage then when you apply a little pressure to release it it breaks under stress... Now I have used the tab next to the one thats broken and drilled a hole in it and used a coat hanger for the pin it will work doing it this way

Dispenser repair kit 8201756. Its available for about $71 including shipping from appliancepartspros.com. I ordered it. It works perfectly well!
Here are my suggestions:
1. You don't have to replace the trap door assembly. Just use the old assembly but you can easily clip in the new door itself, retaining the old pin and springs etc. This is faster.

2. Pry open the outer frame, by sliding a large flat head screw driver in the two tabs at the bottom of the frame.

3. Remove the wires and screws in order. I didn't need any instructions. If you look carefully, you can work with the screws and wires/clips.

Here are the specific instructions for installing the mod kit:

4. break off the broken tab (which had a hole in it in the past) on the right. Not necessarily to make it flat.. just get rid of as much of it as possible.
5. Do a "trial run" of how the metal assembly will seat. Do it 3 or 4 times.. just so that you know how to get it into place easily. While doing this, use the extra shaft pin that is in the kit to align the left tab's hole (the unbroken hole would be on your left as you face the frig) with the left hole in the metal bracket. This is important. This way, you will get perfect alignment for the trap door. Others have had problem with seating. If you do this my way, and actually align the tab hole and the metal bracket's left hole, you will find that the door will sit perfectly.

Now figure out how to do this alignment, a few times, before you peel the sticker from the glue.
Clean the area to be glued with Alcohol and let it dry. This will ensure ears of good adhesion.

Notice that in the instructions, it says don't peel the BACK sticker. So, DON'T. Just the top white stickers and that one stickers that is on the top and also on the back (around the corner). If you don't peel the back sticker, it would be easier to align the pin as in 5. Now, with the stickers peeled, use the pin to align the hole of the left tab and the left hole in the bracket and now press it up, to make the glue stick.

Now you can assemble the rest..

This worked well for me. Thanks to this forum, I was able to find this kit. I had called Whirlpool and Kenmore (mine is a Kenmore) but they didn't have a clue about this fix! IDIOTS! Google and forums like this are always much more useful than the manufacturers!