Invade Mexico? And then what? Every Mexican who is able will be stampeding north straight for the border. Build a wall, not a fence and keep them out.
Fire every judge who has any suspicion of wrongdoing.
Fire every law enforcement officer who has any suspicion of wrongdoing.
Set up a few base camps from which the United States may launch 'Kinetic Military Actions' against drug gangs.
Oversee fair elections.
Stay for a while and wring the damned drug gangs out of that country.
It would be a much tougher go than Afganistan and Iraq, but the payoff would be higher.