Music Hall, Orange, New Jersey
The Mark Twain and George W. Cable speech did indeed take place at the Music Hall located in Orange, NJ. I'm attaching some files from the East Orange Gazette from 1884 that feature advertisements and brief articles regarding the lecture. An 1887 business directory sets the Music Hall's location at the corner of Main and N. Day Street. Subsequent names for the hall are 1908 - Orange Theatre and 1920 - Bijou Theatre both with the address of 243 Main Street
The Music Hall was designed and built in 1880 by architecture firm Silliman & Farnsworth (picture and source attached).
Also a search within the free google e-book: The Founders and Builders of the Oranges: Comprising a History of the Outlying District of Newark, Subsequently Known as Orange, and of the Later Internal Divisions, Viz.: South Orange, West Orange, and East Orange, 1666-1896 by Henry Whittemore has a section about the building of the Music Hall.
A librarian named Alice McMillian may be of more help at Orange Public Library.
East Orange Public Library - through Facebook