So, what book contained the first official history of Jehovah's Witnesses? No, it was not Jehovah's Witnesses in the Divine Purpose (1959). It was Qualified to be Ministers published in 1955. Here is a file that contains the section "Modern History of Jehovah's Witnesses" taken from that book. The chapters are apparently a condensation of a series of articles that appeared in the Watchtower that year.
The file (only about 2.5 MB) can be downloaded from this filesharing site:
http://www.filesend.net/download.php?f=e1ab22e034144e49770f5b4d0e1046bd
I'll leave comments on the material in this first official history to others.