Cardboard and mulch won't work, you need to use Round-up for mint. Use a sponge mop to apply to the entire area, wait 2 weeks, reapply to any regrowth.

Mint, "Mentha", if not contained can take over a garden and some now consider "Mentha" to be invasive. You could use any of the glyphosate products, which might knock it down for a short time but will require repeated applications for years, or you could dig it out and be rid of it. The glyphosates are not as environmentally friendly as the manufacturers would have you believe and researchers have found wee critters where it is used that have genetic variations that cannot be attributed to the natural process. Things such as 5 legs, both sexes in the same wee critter, etc.

If one can smother Canada thistle with cardboard, no way mint can break through it. As long as there is enough weight on the cardboard, that's the key. It has to make solid contact with the ground, and have overlapping edges. If there is space under there, or cracks, it will come through. A few rocks here'n'there, or straw won't cut it. You need at least 3" of mulch or equivalent weight to make it work. Blocking the light is also part of the process, as well as the inches of covering material adding enough weight to cause the cardboard to conform to every contour of the soil surface. Very few plants are strong enough to break through this, definitely not any mint. It's not bamboo.