My two cents would be:
1. I'd take joe's advice and, for a start, take down the stained board.
2. On a dry day, get out there and clean/wipe the gutter.
3. Start lightly water testing from the outlet up the fall. Short sections of gutter. If the leak is in the gutter this testing will pin-point it.
4. Keep going inside to observe - take your time testing and observing.
5. When a suspect point/area is located, then i'd take xenon's suggestion and "clean, prime and coat ... " etc. AAMOI, zinc can be glued with waterproof epoxy.
6. hardmetal's analysis of the defect/history ( lead to zinc with incorrect substrate for one piece zinc) is spot on, but to refurbish would be expensive, and no matter how good i'm sure his repair would be, it should still be tested and observed from below. Hence, the p/b removal.
note: remove the brick and any debris from the parapet and secure the flopping tv aerial.
The d/h window frame and trim req's attention.