I am sorry you have inherited the mint. It is very likely it will outlive you and I. Go ahead and plant your peonies, just be resigned to a lifelong battle with the mint. Peonies are able to grow roots that will do well if you don't give up pulling out the mint as you find it. At least you hands will have a nice minty flavor. Al

Here is another use for the digging fork. If you have such a tool, use it to rout out as many of the roots as you can: in the case of mint, just pulling the top growth won't get you very far. Or employ a shovel, though it will chop through the roots. Work the area thoroughly before planting anything. Probably you'll leave some mint behind, but the more you remove now the happier you'll be in the future. And remember to keep pulling: this is a war of attrition.
Melissa

FYI, I have lived here for 43 years and am still fighting mint my neighbor planted over 50 years ago. For heaven's sake round-up it first before you plant anything, or you will fight it all your life.

I always find mint fights funny since I have been trying for 17+ years to get apple mint to live more than a couple of years in my soil. I generally keep mints in large pots except for the apple mint which attracts all kinds of pollenators but no luck with it. Apple mint contained itself at our old house without taking over the world but no luck here. Keep up the fight Digging up all the segments is the best idea. On other plants that take over from the roots I allow them to dry then put in an old weber kettle and set on fire. Depending on what they are I sometimes use the smoke but normally just let burn.

Thank you everyone for your suggestions! I'll start off with round up followed by pulling and digging out whatever I can. If that doesn't work... I may just have to get use to mint-scented peonies!