The Kansas State Historical Society at http://www.kshs.org/ holds one of the largest newspaper collections in the United States. They have almost every newspaper that was printed in Kansas (well over 61,000 volumes) and more than 12,000 out-of-state-volumes. These papers are in the Newspaper and Census Division, and many are on microfilm. The Family History Library has not collected copies of Kansas newspapers.
A useful guide to Kansas materials is History of Kansas Newspapers: 1854-1916 (Topeka, Kansas: Kansas State Historical Society, 1916; FHL book 978.1 B3k; film 1036378).