Your suggestion of a "Korean problem" assumes that something needs to be done about North Korea.
Setting aside the loose cannon in the White House, what should any administration do? Anything?
By now it is beyond unrealistic, it is pure fantasy to imagine North Korea relinquishing its highest technical/industrial achievement, functional fission bombs. Is this a problem?
How has the possession of a nuclear force helped other states? Has it helped Russia coerce its neighbors, or the rest of the world? China? For that matter, has our own overwhelming and provably reliable and astonishingly accurate nuclear force aided our foreign policy?
I feel there is no real argument for the positive value of nukes. That is, I feel there is no evidence that possession, in a world that has other nuclear states, leads to any particular advantages. It seems obvious that it is a real deterrent to major military threat. No nuclear-armed state has been threatened with invasion, even by other nuclear states. Worth remembering that the only use of nuclear weapons was against a non-nuclear state, with no other nukes anywhere in the world.
There are strong logical arguments that the ultimate weapon does deter, does protect territorial integrity. And there is historical evidence that this is the case.
The risk of active weapons is therefore not intentional use, but accidental use, or a crisis getting out of restraining bounds. In both cases, the historical record shows scary accidents, false alarms, and a series of showdowns between the U.S. and Russia (as well as India and Pakistan) in which caution held.
I feel that the reality of nuclear weapons impresses everyone who is involved with them, that all who have roles in that world take them more seriously than anything we can imagine. It seems that some extra level of sobriety overcomes all who come near the brink.
Has North Korea tried to invade and conquer South Korea since the truce? Why would they do so now? The deep question is why does North Korea need a strong military and a threatening weapons deployment? The two are not identical, but related.
Is it hard to accept that NK feels existential threats? When the West (mainly U.S.) has been talking of overthrowing the government for several decades, it is only sensible to feel scared, and to take steps to deter action on that goal. The best way for the South Koreans and allies to invade NK is across the border, of course, so that is NK's defensive line. The artillery that threatens Seoul is the same that defends against invasion.
That defensive emplacement doesn't work for bombing, so anti-aircraft defenses are needed. But those would not help against submarine-launched nukes---only an equivalent response, a nuclear deterrent, is sufficient for that.
So let's now get back to the main question: Is there anything to be done? Because this question requires a need for action, I went through the issues regarding force and deterrence, and state stability. I do not see an imminent threat---there is no indication NK has designs on neighboring territory, or has intent to attack U.S. interests anywhere. I see only a preference, on our part, that no other states join the nuclear club, preserving our dominance.
But that dominance is not really useful, except in the simple sense of deterrence. I doubt any state, or population, believes the U.S. would simply attack another state with nukes because they didn't act as we wished (preemptive conventional invasions aside). Put another way, no one is scared of our nukes, unless we go crazy (hmm).
The same argument gets used against NK, because otherwise there is nothing to worry about. If NK simply has a nuclear deterrent, so what? Only if we felt we had some inherent right to invade at a time and place of our choosing would that matter (hmm). It is only by characterizing uncooperative dictators as "madmen" that we can justify coercing them and threatening them.
I was not even bothered by Iran's desire to join the club of high technology. Since Hiroshima and Nagasaki, no one has found any coercive value in possessing nukes, but all have found security from their deterrent effect.
Finally, no state that has achieved functional nuclear weapons will ever give them up. Only South Africa has publicly relinquished them, and there is no sense that they had gotten to a deployable design. By now, efforts to force NK to give them up will fail. There is no way we should expect a proud people to yield to our desire that they expose themselves to our coercion, that they throw away their national accomplishment of the highest industrial art, and that they change their government to please us.
Realism is realistic. As Obama said, "Look at the world through their eyes". North Korea has traveled alone for decades, and is not persuaded by our arguments. They will accept assistance that carries a price when they are desperate---I don't feel they are desperate right now. I feel they are confident they have made it past the barrier to membership in the group of nuclear-armed states. Time to get used to it and work out some security agreements.
The main thing NK wants is to not be threatened with military action for the purpose of overthrowing their state. What they do not want is to commit suicide by attacking anyone who would respond with overwhelming force.