I would guess that he did not have a middle name.
The North Dakota Centennial Blue Book, lists all of ND's governor's and their terms. For the majority of the men, a middle initial is listed, but there is none listed for Langer. When I look at the official subject headings of the biographies or political overiews for Langer, what librarians call the "authority file"--the official form of the name used consistently in subject headings is simply "Langer, William, 1886-1959."
None of the books my library has on Langer (
The Fighting Governor, Dakota Maverick, Unequal Contest:Bill Langer and his Political Enemies) mention a middle name.
Beyond this, I would check with
the Institute for Regional Studies, which is a archive/history library within the North Dakota State University Libraries. Their staff should be able to verify it if anyone can.