Why not Use the Word "Whose" Here?

Either "Whose" or "whom" would be a correct term to use as the sentence is currently constructed. However if it said "Mr.Shaikh, who, family members said, had a history of mental illness." Then "who" would be okay.

I understand what the author was saying, but I can see why you could be confused. He's saying that the family members said he had a history of mental illness. Think about it without the family members for a minute. What if it said "....to Mr. Shaikh, who had a mental illness". I agree that adding "family members said" just makes the sentence more confusing. I had to read your question a few times before I could understand what the author was saying also. I hope my answer is helpful. In my head I understand what you're asking and what the author is saying, but it's kind of difficult to type the answer clearly in a way someone else can understand...