The dry wall immediately next to my shower tiles was damaged by being exposed to too much water over time. I let the wall dry out then removed the damaged section of drywall by pulling it out with my fingers. Now there is a hole/gap between the tile and the undamaged drywall (about a 1 inch wide gap that is about 3 inches tall). I have patched drywall before with a patch kit but I am not sure how to go about fixing this gap. There is nothing but air behind the hole and I can't attach anything to the tile.
Take a paint stir stick from a home improvement center, break it into pieces that will fit behind the patch hole, and pop in a screw to hold it in place, on the side away from the tile. It doesn't have to be rock solid, it's purpose is to keep the drywall patch in place long enough for the compound to dry. Once the patch is in place (liberally buttered with compound) and dry, you can remove the screw holding the stick in place- the stick may drop off into the wall, but that's no great problem. Patch the screw hole, and you're set.
You might have some luck with a wooden brace attached to the drywall above and below, and then patch in a small piece of new drywall, attaching it to the brace. You can spackle the gaps, and then re-caulk against the tile.