Honestly, Trump's win IOTL kind of ruined any requisite for holding office before that. I was thinking this...
Jon Stewart at the time of his retirement was already popular. Perhaps realizing that Donald Trump is becoming front-runner, Stewart decides to use his fame for good and launches on his finale a campaign for the Democratic nomination for President of the United States. At first (Like Trump's) his campaign is treated as a joke by the media, but unlike Bernie Sanders has a really good following among young people, and is more relatable compared to the other two major candidates, as he has school-aged children. Although he is outspoken, his views are less radical than Sanders' and he lacks the scandals that Clinton has. The primaries are a tough battle, but after a few debate performances he surpasses O'Malley and Sanders and goes head-to-head with Clinton. Going all the way to California, he ekes out a victory against her, winning the nomination.
Selecting Amy Klobuchar as his running mate, Stewart goes up against Trump. The comedian against the reality star, and Stewart with his sharp tongue and humor manages to win over the American people. Showcased are his charitable acts and general goodness, and without Trump surrogates being able to pull the "But Hillary Clinton..." style attacks on him, he manages to perform well. He ekes out a bare victory over Trump in November, winning Florida, Michigan, Wisconsin, and Pennsylvania.