Ike's health was too frail, he's not going to run in 1960 anyway,
Reagan will most likely run in 1988 and easily win, presiding over the fall of Communism and the First Gulf War. He goes out being regarded even more of a national hero than OTL. However, this most likely does not butterfly Clinton away, since undistinguished and relatively unpopular Bush Sr. is still going to lose the election to charismatic newcomer Clinton when annoyance about the recession and incumbent party fatigue hits the the Republicans in 1992.
Clinton will surely run for a third term in 2000 and win, getting a vindication for the impeachment and setting himself as the historical nemesis of the Bush dynasty. He rallies the nation when the 9/11 attacks hit, presiding over the invasion and post-war reconstruction of Afghanistan. He goes out being regarded as a slightly controversial but overall rather successful President. Since in 2004 the economy is strong, election between Gore and McCain is a toss-up.
Since Bush Jr. will be trounced by Clinton in 2000, his Presidency is butterflied away, thanks the AH gods. Anyway, had he somehow got the Presidency in 2000, if he had run in 2008, he would have caused the worst Republican landslide electoral defeat since 1964 running against Obama. Think of the Democrats easily getting at the very least another 5-6 Senators, 20-25 Representatives, and 400+ votes in the Electoral College (at least Missouri, Georgia, Arizona, Montana, North Dakota, West Virginia go Blue as well). 20% approval rating and widely being regarded as radioactive by your own party spells that. He might likely still get the nomination thanks to the support of the extreme conservative base, but after that, it will be a lemming charge.