From your query I started to look for birth or marriage of a Danith ROSE to a John W COOK. Seems like I was wasting my time as you already have this information below on Danith Rose HANNIGAN (Or Rose Danith HANNIGAN) on a USA board.
Name: John W Cook
Age in 1910: 9
Birth Year: abt 1891
Home in 1910: Renton, King, Washington
Relation to Head of House: Son
Marital Status: Single
Father's Name: John W Cook
Father's Birthplace: England
Mother's Name: Rosa D Cook
Mother's Birthplace: West Virginia < < < < < < < <
John W Cook 40, ENG - vcoal miner
Rosa D Cook 40, WV
John W Cook 9, IOWA - coal miner
Carrie L Cook 14, IOWA
Ronald C Cook 12, IOWA
Charlotte E Cook 10, IOWA
Oscar L Cook 8, WASH
Charles G Cook 3, WASH
Herbert G Cook 2, WASH
So what you are looking for is someone called John W COOK born about 1870 somewhere in England who may have never shown in any UK census.
Obviously he was married in USA. Does his marriage certificate give any information on his parents or have you located him on any earlier USA census?
You could use FeeBMD to see how many people called John W COOK were born in England between say 1868 and 1872.
FreeBMD has the civil registrations from when they started in July 1837 up to around 1940/50. http://www.freebmd.org.uk/
You can order certificates from here for £9.25 and they will arrive in about a week:http://www.gro.gov.uk/gro/content/certificates/