Do not becomes don’t, cannot becomes can’t, and shall not becomes shan’t. Won’t does not follow the same pattern.
That is because won’t is actually a contraction of woll not. Woll is an archaic form of will; many Germanic languages have or had a similar word with a similar meaning.
Won’t fought off competition from other forms including wonnot, woonnot and wo’nt to become the standard contraction we use today.
We may no longer use woll, but it is easy to see why English has retained won’t instead of using willn’t or even win’t.