I have no knowledge of Karen's data, but the church records are now online and free in the Danish Archives site: http://www.sa.dk/ao/
English instructions for AO: http://www.utilis.dk/ressourcer/2011_Generic_instructions_fo...
With the info you listed their records were easy to find, as Amly's birth record lists her parents' marriage date and place, and the birth records also list the family's address in Rønne which simplified finding them in the censuses.
Do you know if the brother Svend also came to USA?
Here's an exact spelling and birthdate match, but coincidence is possible:
Social Security Death Index
Name: Svend Pedersen
Last Residence: 60622 Chicago, Cook, Illinois, United States of America
Born: 21 Feb 1899
Died: Nov 1973
State (Year) SSN issued: Illinois (Before 1951)