I would start with a gutter company, since it could be leaking back behind the fascia board into a soffit or whatever the design was used to run a gutter between two roofs. Have them direct you as to a roof issue or lanai issue.
I just did a quick search on this site for gutters. Found multiple post talking about who installed and/or repaired theirs.
I have not needed one and can't advise on who.
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