How to fix the issues with this roof

As you can see, both corners of the chimney have a decent amount of water damage to the sheetrock. 

The drip edge is installed behind the gutter. This allows water to flow behind the gutter, on the fascia, and down to the soffit.

Here is a picture of the soffit. The red section with an arrow is rotted severely, and the yellow team with the hand is how much of the wood we will have to cut out to repair this area.

This chimney flashing has a few issues. The main point is the hole in the caulking from it shrinking over the years. The other problem is the flashing is open on the ends. Not to mention a few large openings in the mortar.

The end of the flashing is wide open.

No mortar, and it is wide open!