There was no massive voter fraud in Illinois. There was *some* fraud in Illinois, by both sides, but nothing out of the ordinary. "Illinois stolen in 1960" is an urban legend, in part a consciously fabricated one.
When you get to heaven one day (not anytime soon, I trust), you'll find out that Nixon actually carried Illinois in 1960 by 1,328 votes. You'll also learn that Kennedy similarly lost Missouri by 845 votes and barely carried Texas by 3, 205 votes.
Wish I could be there to see the look on your face, but I'm guessing I have reservations in warmer climes.
I think Nixon only won Missouri by 700 votes.