I did some additional attempts on Alice and her children in Illinois. Also some on the McHenry "potential". Nothing found of concrete nature.
Since the correct Alice was born in Illinois, then was there for the birth of Leslie in 1912/13, I'd suggest working that area first. Attempt to obtain a birth certificate of Leslie (unless Illinois law has a 100 year prohibition).
I tried to find them in the 1930 census; however it was not transcribed to identify birth place of parents, so much larger lists of criteria facts are given, which requires individual review of images. For example, the 1930 Census has 84 "Alice" born 2 years of 1872/1873, but that was only a listing for Chicago, Cook County, IL. If I could have directed a search using parents State of birth, that would have helped.
There is microfilm for the birth years of Illinois 1872/73; also for 1912/13. There are microfilms for the death indices. I can't find suitable online databases that might help in the research, although I had searched death indices for the name of Stoltz, McHenry, Cornell, without success.
Volunteers are a plenty for Cook County, IL on the Random Acts of Genealogical Kindness. READ the guidelines first:http://www.raogk.org/illinois.htm
Although it indicates links for Jewish Genealogy, it is a great listing of indices for all. It will give you some ideas:http://www.jewishgen.org/Infofiles/Chicago/private_orgs_chgo...
I'd suggest the usual birth, death, obit searches. Give the volunteer the information you have.