The family settled in Jackson County, Iowa. It sounds like Anna (Nagle) Miller who came to US in 1885 d.in 1888 might have had just enough time to do the declaration of intent, but there might not have been a rush. I might be able to go to the National Archives to find for Iowa. Or, perhaps the county courthouse.
Perhaps I should aim for ship records? to find out if Joseph Miller (Mueller) travelled with the Nagle family in 1882 when I believe they came. Since there was no family stories about Joseph Miller - and doesn't seem to be any burial etc near Anna perhaps he never made it?
In 1882 Louis Nagle Sr.came with wife Corda or K. and children Louis Jr. b. 1863 & Bertha Nagle (b. 1872?) (married Andrew Morey). Bertha Died 1895.