In my opinion, the only real way to deal with them now is to use an EMP type weapon to take out the south facing artillery/missile sites and runways, then physically destroy them with a coordinated guided missile/artillery and air strikes. Is it any surprise that the US are currently developing such systems?
http://www.dailymail.co.uk/sciencet...developing-megapulse-weapon-cripple-city.html
Air power is another question mark, their air force has been conspicuous in it's absence recently. It's likely they do not have the capability or parts to maintain the ageing craft, and as such, it's likely to be a non-problem. Especially as the AFH of NK pilots are very low! They are not exactly going to be able to compare with US pilots with significantly higher AFH. If they even manage to get any aircraft airborne at all...
It's the only way to make it work. However, then the real problems start. The only reason China prop up the dynasty is that it keeps millions of refugees behind the border. NK also acts as a nice land buffer between it and the American allies of the South. If China end up having to prop up what remains of NK, its going to make them less than happy.
Politically what happens then is well out of my limited and uninformed ability to even begin to predict, but I think the appropriate word would be "instability".
It should be noted that Any such military plan will cause civilian casualties in the south inevitably, and possibly more further afield. The odd artillery shell will get through to SK, maybe more, provided the artillery are well trained enough to operate the guns manually.. Which they likely will be, for obvious reasons. Also, we don't know what (if any) reliable SLBM systems they have, which really throws a spanner in the works. Once they have that capability for submarine launched ICBM with a true nuclear warhead, they become even harder to deal with.
Again, in my opinion, now is really the last window of opportunity to "deal with the NK problem" militarily before they become almost impossible to deal with safely. Even now, it might be too late.