I began methodically researching the names of all the PERKINS children, even the youngest who might not even be found on the 1900 Census.
Usually, I do not find the females, as they marry and it is a lost cause unless you happen to be in their direct line, or have information about who they married.
However, a female is found listed for the children of John and Emily, with both parents' surnames also listed, including the mother's maiden name.
Unfortunately the town does not exist as it is written, for her birth:Littlefield.But there is a place called Litchfield, in Montgomery County, and this child was born in 1898, a mere two years before the census.
I know it is only a hunch, but could I persuade you to look at Montgomery County, Illinois, for a JOHN PERKINS (KY-1866) and EMILY (IL-1868)??
Once again, the children's names are Wyatt, Ethel, Roy, Judy, Horace, Harriet, Evelyn, Marie, George, and Frank.