Here is where I purchase my Eternabond tape. I have used it for many years and it is the best there is. I did every seam and penetration on my trailers roof about 12 or 13 years ago and it is still holding up like new.
For your application, I think I would get a 50 ft roll of 4 or 6 inch wide tape. Clean the area well, glue down or cut off the loose stuff, and then put the Eternabond down starting from towards the center of the roof and work your way to and around the edge down to the gutter.
When done, go ever the exposed edges with
Dicor self leveling sealant and you should be good to go.
Barney