No, it is illogical to nuke North Korea. Nuclear warfare is so intense and improved that simply nuking a country could affect neighbouring countries to the extent of eradication of human populations. It is a chain reaction.
Starting off the second round I would like to present the case of the two atomic bombings of Hiroshima and Nagasaki in the year 1945. These devastations have made a significant mark and aftermath that is still present today. I would lie to point out that, the attack had shocked the world and had taken out two major cities in Japan. This attack was only afflicted by an atom bomb, a more minor nucleic weapon. Present day nucleic weaponry consist of arsenals of pure destruction that are ten times more powerful that the Nagasaki and Hiroshima atom bombs combined. In addition to that, the aftermath is more horrific as future generations would also be affected.