Assuming that Ike's Health Problems are less Severe, there is no 22, and he actually wants to run again for some reason then I could see this come into fruition. Ike was a very popular president and even though JFK would put up a fight, everybody likes Ike so he wins comfortably but it is no 1952/56 type margin. JFK does better in all regions slightly though heavily better in the south with Ike being much more moderate (and wouldn't be held so highly after his integration attempts by the white establishment) then Nixon and probally would never try to appeal to white southerners like Nixon did so loses some ground from 1952/56 though still wins most of the Upper South and Texas. Though if this really happened then i would see a southern third party rise.
Dwight Eisenhower - 351 52.02% John F. Kennedy - 186 46.48%