This popular vote for United States Senator was technically only an advisory vote as prior to the adoption of the 17th Amendment in 1913, state legislatures officially elected United States Senators. The Nevada Legislature ratified King's election on January 27, 1909.
The primary source is the official manuscript returns in the 1898-1918 poll book, located at the Nevada State Archives in Carson City. The results were also printed in the "Official Returns of the Election of November, 1908", but this report contains a misprint for Lutz in Humboldt County.
Source for Popular Vote data: Nevada State Archives, "Choice for U. S. Senator," Election Returns of General Election held on the third day of November, A.D. 1908 (Carson City, 1908)