Sorry. Arkham Asylum was actually built like 150 years ago. Its mentioned in several of the Cyrus Pinkney files and its history is told in the Ghost of Arkham files found in the first game. There are several references to it in the Blackmail files in Origins and Quincy Sharp even talks about opening it back up.
Ok. That's the part that led me to believe that this was the story of Arkham Origins.