Q:

Has an AI ever solved a detective mystery? 

In detective novels, the point is often that the reader gets enough information to solve the crime themselves.  This "puzzle" aspect of detective novels is part of the attraction. 
Often the difficulty for humans is to keep track of all the variables - events, items, motivations. 
An AI would have an easier time keeping track of all the details, but would rely on real-world knowledge to prevent making crazy mistakes.  For example, if it was stated that a character took the train, the AI would need to know that this is a method of transportation - that it changes the location property of an agent over time. 
Has an AI ever been able to solve a detective mystery? 

A:

I might be wrong, but I do not believe that something of the scope you describe would be possible with the current state of technology.  It would require a lot of things which are still in relatively early stages of research. 
For one, just extracting relevant information from text is a huge task by itself.  Doubly so with a novel which contains a large amount of unimportant details. 
It might perhaps be easier if the input was presented in the form of some sort of list of important facts.  But it would still be rather difficult for the AI to connect them and find a solution. 
As an example, let's say that we have these two facts:

Alice died of a snake bite. 
Bob was seen buying a couple of mice recently. 

To a human, it seems obvious that the mice were bought to feed a venomous snake.  However, it would probably require a tremendous effort to teach an AI to make such connections. 
Disclaimer: I don't do text processing myself, so I'm not quite up-to-date on the current state-of-the-art.  It's possible that some of these things have already been done in some form.  If anyone knows more about the subject, please correct me if I'm wrong.