I see that the 1940 census has already been posted for you, showing that Else Freeman was living with her daughter Mildred at that time. If she had a younger daughter, born ca 1932-34, that child should be living with her in 1940.
Apparently your father did not follow up at the time the daughter was mentioned, circa mid-1930s? He did not verify that the story was true?
There are three possibilities I can think of:
1. Else had a daughter by your father and gave her up for adoption (or at any rate placed her in another home). Else was a single mother when she gave birth, with sole responsibility for a twelve-year-old, and may have felt she could handle no more.
2. Else had a daughter by your father and the daughter died before 1940.
3. Your father's friend who said Else had a daughter spoke true, but he/she was referring to Mildred. If the friend had only heard a reference to a daughter (and did not personally meet the child), the friend could have wrongly assumed the little girl was younger than was actually the case and was John's child.
Are you convinced that option 1. is the only option? If so, one would check adoption agencies in Kansas City and environs, and I presume you've done that (as you seem to have been thorough).
If the child died, we would most likely find a death certificate under one of the surnames, but we have not.
Mildred is also a half sister and would presumably have news of this event. Have you found Mildred and spoken with her about it?
Other possible contacts would be Else's siblings. Most people would know if their sister had a baby who died; perhaps they are somewhat less likely to know of the daughter if she was given up for adoption.
Have you researched Else's siblings? Do you know their names and locations in 1940 and onward?
Should the forum help to find Mildred and help to find Else's siblings?