Don't worry, you live in a world where a man with a well documented decades-long laundry list of scandals and faux-pas can brag on tape about how he can get away with molesting young women because of how rich he is and still get elected President.
I'm actually wondering what you could effectively blackmail an American politician with, given how little Americans seem to care about the morality of their leaders. Maybe just the eponymous "dead girl or live boy," but then you have the Kennedys...
Well, in general anything that would get them prosecuted for a felony should be an effective stick. Add in a carrot and now they're on the hook for bribery too.
Well lucky you, we have the exact 100% polar opposite of that system currently. I think it would be fine to move toward it a little.
I'm skeptical of the notion that outing public officials (when the blackmail material is personal in nature as posited by the GP) would move toward that goal.
I don't even think an exception should be made for material that would expose hypocrisy in policy-making (eg. 'pro-life' lawmakers that have had or facilitated an abortion).
Straight-up crimes though (bribery, corruption, theft, insider trading, etc.), sure.