How to properly re-caulk this shower?
If it's breaking up there,it's probably breaking up elsewhere. Think about doing the whole shower, it looks like there's more discolored caulk. Stuff some paper towel down in that large space to see if any water remains and to soak up any that's still there. Once you clean it up and spray your mold killer and let it thoroughly dry, get some epoxy grout and fill in the whole seam between the floor and wall. Normal grout wouldn't be used for a change in plane but epoxy grout works well here. It's waterproof and doesn't need to be sealed.