Known as the Myers Opera House. 118 E. Milwaukee Street, Janesville, WI 53545 

The Myers Theater was built in 1870 as the Myers Opera House. It started showing movies around 1929. In 1977 the Myers Theater was demolished and replaced with a bank. The history page for this theater gives Milwaukee Avenue and South Parker Drive as its location. Google Maps returns East Milwaukee Street and South Parker Drive.  http://cinematreasures.org/theaters/14142 

January 20, 1885

I don't have documentation that this is where the Twain-Cable speaking engagement occurred but the Twain Speaking Engagements site mentions it occurred in the Methodist Church and Janet C. Olson Assistant University Archivist at Northwestern University Library provided me with the address: 

Burlington Opera House

The night the opera first came to town 

The Grand Theatre was designed by Merle F. Baker and was constructed on the foundation of the Keokuk Opera House (Circa 1880) which burned in 1923. It was patterned after theaters in Chicago and was praised as one of the finest theaters in the country at the time.
http://www.keokukiowatourism.org/theatre.htm 

It Happened In Keokuk  Friday, December 7, 1923  

KEOKUK'S 43 YEAR OLD OPERA HOUSE IS GUTTED IN EARLY MORNING BLAZE. 

I have no direct information that the Opera House was the location of the Twain-Cable reading of January 13, 1885, but I found this listing in the Hannibal City Directory 1885-86.  Hannibal Opera House Co., cor of 5th and Center, J.B. Price, manager, office at F. & M. bank.

 Image:  Hannibal, Missouri: Bluff City Memories  By Steve Chou

Doerr's Opera House operated out of the second floor of the building on the northwest corner of 6th and Maine Streets.
http://www.idaillinois.org/cdm/ref/collection/qpl/id/1589

The St. Louis Mercantile Library, founded in 1846 in St. Louis, Missouri, was originally established as a subscription library, and is the oldest extant library west of the Mississippi River. Since 1998 the library has been housed at the University of Missouri-St. Louis. It has 600 feet (180 m) of papers, ledgers, and printed materials currently in 26 departmental or other record groups In 1986 the library received a grant from the National Endowment for the Humanities because of the collection's cultural importance.

January 8, 1885 

From http://sangamoncountyhistory.org/wp/?p=2456  Chatterton Opera House, Posted on October 30, 2013 by editor 

However, on March 17, 1876, Rudolph’s was almost completely destroyed by fire. “There is a story that Mr. Bunn, upon being awakened with the news that the Opera house was burning down, remarked that he couldn’t put it out, and turned over and went to sleep again,” Gib Bunn wrote. 

This church has owned and occupied three houses of worship: The original Plymouth Church, northwest corner of Meridian street and Monument Place, now a part of the English Hotel; the second Plymouth Church, on the southeast corner of Meridian and New York streets, on ground now occupied by the Federal Building, and the third, on Central avenue, at Fourteenth street, which was acquired by purchase and remodeled.  http://scholarworks.iu.edu/journals/index.php/imh/article/view/5761/5242