It looks like the name was Kurvink, based on later records. In 1880, there is a William Rummins in Humboldt County Iowa, living with his son Benjamin, and he is listed as divorced. Some of the Kurvink children are still listed in Iowa, it looks like they ended up in Pipestone County Minnesota. You might check to see if William and Jane married in Iowa, they must have married shortly before 1870, as there was a 2 year old Kurvink child in that year.
In 1860 it appears the Kurvink family (spelled as Curvink) was in Sheboygan Wisconsin, and William was in Washington County.