The Justice Department could prosecute Kellyanne Conway. Or they can use their discretion not to do so. Jeff Sessions is the one who ultimately controls that decision, although Donald Trump does have the ability to fire him if he doesn't like the decision.
The only practical way to punish a president is to impeach him. That's a difficult step and seems unlikely for trivial offenses like saying that a store is foolish for removing his daughter's clothing line.
Note for example that Bill Clinton was guilty of perjury but his impeachment failed. We know that he was legally guilty, as he was disbarred for it. But he wasn't removed from office and never faced criminal prosecution for it. While technically illegal, most people felt that lying about one's mistress was not an impeachable offense--even under oath.
In order to punish Trump, the offense will have to be serious enough that enough of his supporters will peel away and allow Republican politicians to impeach him. As is, his supporters continue to like him and view these charges as just sour grapes by the people who lost the election.
Sending men to break into your opponents' headquarters to look for files was serious. People could point to that as a real crime. Saying that it was unfair for a store to remove a clothing line? Not a real crime. Making a big deal about it hurts your cause, because if people are used to tuning out when a big deal is made, the serious crimes will be ignored too.