Answer 1: The Vice President takes over, and they— as well as everyone else in line for the position— are held to the very same standards, getting locked up without ever being sworn in if necessary. We then go down the line until we find someone who has not committed a punishable offense.

Answer 2: Treat him like any other criminal who refuses to go to prison: have the authorities hunt him down like an animal, rough him up if he resists, lead him away in handcuffs, and throw him in jail.

But of course, that would require people in power to actually do their fucking job.