Why won't the U.S. sign a peace treaty with North Korea? Is there a reason why it wasn't done sometimes after the cease fire? What was the purpose of not signing a peace treaty with North Korea if the U.S. was no longer interested in invading North Korea?
The significant misconception is that, from the information I've gathered, the US was never formally at war with North Korea. It and the United Nations intervened in the Korean War on South Korea's behalf to prevent the North from wiping it off the map. There is, therefore, no need for the US to sign a peace treaty with North Korea because a state of war never existed between them.
Indeed, North Korea and South Korea have not signed a peace treaty with one another, and thus the Korean War is technically still ongoing, but that's a separate question.
As for why North and South Korea haven't signed a peace treaty with each other, two countries not engaging in active war doesn't mean that they're actually friendly towards each other, see https://en.wikipedia.org/wiki/List_of_border_incidents_involving_North_and_South_Korea
The two countries signing a peace treaty would mean that the Armistice would be nullified, and the Demilitarized Zone would cease to exist. Neither Korea wants this, as the DMZ protects both Koreas from being invaded by the other (it especially protects South Korea from being invaded by North Korea, as the North is much less predictable than the South).