The Republic was declared between Oct and Dec. 1963. The date you are using is the date the United Kingdom recognized that independence. The declaration came a year earlier.
Constitutional Status of Islam(ic Law): The Constitution was adopted on 12th December 1963, and has been amended several times; most notably in 1964 when Kenya became a Republic and in 1991 when a multiparty system was restored. The Constitution does not provide for any official state religion. Article 66(1) to (5) provides for the establishment of Kadhis’ Courts
http://www.law.emory.edu/ifl/legal/kenya.htm