The NORKs have an army, stronger than what they had for the Korean War. South Korea's major city and capital is so close to the NORK border as to be withing artillery range. The place is run by a pudgy dictator who appears to be a crazy man. They are building nukes and the ballistic missiles to deliver them. They are so cold stony broke that they are having trouble feeding their people. They have little to no international trade, travel, or connections.
We want them to drop their nuclear program. They are dead set on getting nukes for the international respect (outright fear) that a nuclear weapons state commands. Their nuclear facilities are well dispersed and underground, probably proof against airstrikes.
We would like the Chinese to cut off their crucial imports of food and fuel to force them to drop the nuclear program. The Chinese could do this, they are the only source of supply for the NORKs. Trouble is, the Chinese don't want to squeeze the NORKs that hard, for fear the regime might collapse. If or when that happens, enormous attractive political forces will try to pull North and South Korea back together. A lot of South Koreans still have kin in North Korea who they would insist on saving. If that happens, South Korea, with an economy so advanced it can export automobiles to North America and make state of the art semiconductors, plus an educated population, will run the show, just like West Germany ran the show when East and West Germany reunited. The result would be a capitalist, successful, pushy, Korea running right up to the Chinese border at the Yalu river. The Chinese hate this idea. Especially as the Koreans are so tight with the Americans. I doubt that the Chinese will push the NORKs very hard, certainly not hard enough to gt them out of the nuke business.
Maybe we could get the Chinese to repatriate North Korean refugees to South Korea instead of handing them back to the tender mercies of Kim whats-his face Number 3. Give this a few years to work, and the population loss would hurt the NORKs.
So what's left? We could assassinate Kim whats-his-face Number 3. He deserves it, and the NORK regime would probably collapse into chaos as the various survivors and number two men struggle to take over. And it would probably stop the NORKs from shelling Seoul into rubble. On the other hand, collapsing the NORK regime is scary all around.
We could start up the Korean War again. Nobody likes this idea. For good reasons.
We could shoot down, or shoot up, any more NORK missile launches. There is already newsie speculation that we caused yesterday's launch failure by computer hacking or black magic.