White Lake, South Dakota is one of the many Luxembourger settlements in the American Midwest. According to the Institut grand-ducal, it was founded during the greatest period of Luxembourger migration - 1860-1900. Little information about the area is available online. The only local history I could locate was that of the catholic church, St. Peter's.
However, if you have a little patience, you might find what you need. The local Rootsweb offers lookups in history books:http://www.rootsweb.ancestry.com/~bwo/sdakota.html. The county historical society likely has more records. They do not have a website, but they do have a post office box. Send a letter or email. Their contact information follows:
Aurora County Historical Society
PO Box 632
Plankinton, SD 57368