Using Tremco Instant Patch Roof Repair and Tremco Roof repair fabric.
1. Ensure that inside gutter is clean
2. Lightly sand with 200 grit sand paper
3. Must be dry
4. Apply a brush coat of Instant Patch over seam extending 4 - 6" either way (fig 1)
5. Cut fabric and lay into wet patch overlapping at seams at seams from both sides. (fig 2)
6. Apply second coat of Instant Patch over fabric (fig 3)
7. Warm with hair dryer until cured.
Tremco Instant Patch Roof Repair and Fabric is available at most Building Supply stores, look in the roofing section.
On the referral network there are companies that can repair your gutters for you and often if the gutters are old and corners have deteriorated over the years, it is often better just to change the whole corner.