This is what mine looked like until last weekend:
The Mint plant's roots tend to grow outwards, towards the edge of the pot. The older part of the plant, in the centre, becomes tough and woody.
This is my procedure for re-potting Mint.
Stage 1: Tip the plants out of their pots. You will see straight away the truth of what I was saying about the Mint producing lots of roots:
In this next close-up photo you can see how the horizontally-growing roots put up new shoots at the edge of the pot:
Stage 2: Using an old kitchen knife I cut some "chunks" [root cuttings] off the plant, each with a good amount of fat white root and preferably one or two new young shoots. I use perhaps 5% of the plant:
Stage 3: Half-fill the pots with fresh compost (I always add a layer of stones at the bottom, to aid drainage), and put in a couple of the chunks of plant:
Stage 4: Cover the root cuttings with another layer of compost, leaving any new shoots sticking up above the surface, and water-in:
That's it - job done!
The other 95% of the old plants is placed in a plastic sack for later disposal at the Council tip. Never put the old plants in your compost bin - they would simply take over. Of course, if you are really keen, you could pot-up dozens of little Mint plants into suitable containers and grow them on for selling at a Church Fete or something. The success rate for establishing Mint from root-cuttings is practically 100%, and lots of people would buy a decent young Mint plant for a £1. It's not for nothing that making money is called "minting"!
In a few weeks' time these new Mint plants of mine will be putting up strong new shoots and by about May I should be able to harvest from them. The re-potting task is a mucky one (especially on a cold day), but it is certainly worthwhile.