Trouble with Mint

Trouble with Mint

Backstory: Two years ago, my boyfriend and I bought a cute little house with a small yard. One of the things I loved about it was the cordoned off little patch in the back for an herb garden. However, we had no idea that the garden area had been neglected for years, and furthermore, whoever had originally planted it had installed mint. I realized once spring came around that the mint had taken over, aggressively. I'm generally super busy - two jobs - and don't have tons of time to garden, but I'd really like to use that patch for some harder to kill plants. But I have to get rid of the mint first. I'd rather not use crazy chemicals if that's even possible. I thought about covering the whole area with manure and leaves, and putting a tarp over it to see if I could bake it out over the summer, but would that even work? Any suggestions would be highly appreciated. Thankyou!
before&after: my office

before&after: my office

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