This mint is not dying. It is languishing. What kind of soil is in that pot? (all potted plants should have potting soil medium only, sterilized). The bark on the surface tells me this plant is in garden soil. Way too much soil for the size of plant (small). Does this pot have a drainage hole? How often do you water?
I doubt this mint is root bound yet, looks like too much soil, too few roots and root rot.
I would transplant chunks of this plant in fresh, sterilized, bagged potting soil in 4" diameter pots. Perhaps 3 little clay pots? Use Osmocote 14-14-14 fertilizer at half the recommended amount and applications. Water when the pot feels light. Get used to the heft of a potted plant and its soil when watered. It feels very light when it needs to be watered. Never water every day. Only water when that pot and soil and plant feels light.
Fill the new pots 1/3 full of potting soil, water, then add a chunk of mint plant with roots and then fill around and up to within 1" of the rim. Firm the soil as you fill around the plant. Keep the roots beneath the soil and the rest of the plant above the soil!
Your plants are planted in a too large pot, garden soil(?) and the surface of the soil is too low from the rim of the pot. Easy to over water. Garden soil will not work in pots for potted plants. Hard and fast rule. Let us know if my assumptions are correct?
Also, clean the debris off the surface of the soil for potted plants. Harbors fungus, disease, insects. Not good for potted plants. In the larger body of the garden it works, not for potted plants.