If your immigrant ancestors had ties to the Midwest, you might want to consider looking into the Danish Immigrant Museum and Family History and Genealogy Center. This is a relatively small organization located in Elk Horn, Iowa, with numerous possibilities. Their website alone may hold the keys that can unlock your mystery. You can visit it at http://www.danishmuseum.org/Welcome.html.
Included in their collections are photographs submitted by people with Danish heritage, biographies and histories of Danish immigrants from all over the country, and access to some specialized records created by Danish organizations. They also have a helpful historical timeline that can put things into perspective for you.
When the Danes got settled into the Midwestern states, one of the ways they kept their traditions alive was by forming their own societies. Two of them that sprang up were the Danish Brotherhood and the Danish Sisterhood of America. The former was an insurance company that encompassed several groups that had been organized to promote heritage and was founded in 1881. The records of its lodges include such information as full names, birth information, where their membership moved over the years, etc. These records are invaluable for that area covering Iowa, Minnesota, Nebraska, the Dakotas, and anywhere else there was a significant population of Danes.
The records produced by the organization can be found in different places. The Danish Immigrant Museum has information covering the years 1881 to 1995 on microfilm. Many of these are indexed and can give clues as to origin in Denmark and native name spellings.
Another avenue to search are educational facilities founded by ethnic groups. These institutions would have also kept records of the students who attended, although they may not be held on site. You can contact the institution for direction in finding where the records are kept.
Although these facilities will most often hold information on people who settled the areas, some accept histories and biographies of Danish immigrants in general, so it never hurts to check when all else has failed. Another thing to consider is that although family groups most often settled together, that was not always the case. The Midwest could hold more records for you than you know.