This family really doesn't want to be found! I have searched the 1870-1880-1900 census and cannot find any member of this family--even though they were clearly together in 1870!
I did find marriages in the IL Secretary of State's marriage index for: Nelson Gilmore to Sarah Hardesty November 6, 1864 in Crawford County, IL
Rosa Gilmore to James Nichols September 13, 1883 in Champaign County, IL
The Christopher Gilmore in Vermilion County in the 1910 census was born in Ireland and is living with his brother James Gilmore, also born in Ireland. Christopher is reported age 44, while James is 50. They report immigrating in 1883 (James) and 1889 (Christopher).
Do you have any census information for Nelson and Sarah showing them with their children? Where were they living at the time? Since Rosa is your 2nd great grandmother we know she lived long enough to reproduce.
Good luck with your brick wall! Ellen Fanning Coulter