It seems to me that teaching potential demagogues that they will go unpunished if it's sufficiently inconvenient is a bad lesson to teach if you want to keep a republic.
You can't impeach or have any justice done in two weeks. It's just not worth it.
If there's some kind of crime that's been committed and it can be prosecuted successfully, then by all means after he's out of office. I don't think that's the case though.
Just maintain order and wait it out.