How do I get a cleat that's stuck in a pedal, detached from the shoe, out?
Your goal is to apply torque to the cleat as you would with your foot in a shoe.  So essentially you need something like a gigantic flat screwdriver.   I have a mini crowbar that would do the job, for example, but if you have any metal tyre levers they might be big enough.

Slacken off any tension adjusting screw first