United States would prevail quite easily, however, their criteria for victory in such a circumstance would be similarly high- it may not actually be a 'Victory' in their eyes.
(also, North Korea may hate the United States in writing and the masses, but the leadership is more educated and calm and collected, which, in conjunction with their totalitarian control over the nation SHOULD ensure a reasonable amount of stability - They probably dont actually want war, because war is change, and change of any sort risks destablisation, which is unacceptable to what is basically an aristocracy in North Korea)
That (first paragraph criteria) is to say, North Korea has a ****load of conventional, nuclear and (suspected) biological weaposn at its disposal, all pointed at South Korea and Japan, in the event of Total War (and North Korea, I would daresay, knows no other kind), both countries would be devastated and the United States would lose two key allies in the area.
On the note about China aiding North Korea, I personally do not think so, I do not think they are prepared to go to war with a Nuclear superpower for the sake of one troublesome and self centered (in terms of the leadership class), nor would they risk losing vital trade ties with the United States and her allies that are (currently) her economic lifeblood.
So yeah, in the unlikely event of war, the United States would prevail (though a conventional war with North Korea may result in something that would make the Pacific Campaign of WW2 look like childs play) in either a nuclear war or regular invasion (North Korea's nukes arent advanced enough to reach the US just yet, and the US has ships with sufficient anti-missile technologies to intercept), the United States would 'Prevail', but at a terrible (if not UNACCEPTABLE) cost in terms of the geopoltical balance of power within the area, such as: the loss of allies, the general angst of the neighbours, the radioactive fallout of a nuclear war, the high casualties that would result from an anti insurgency campaign (that is almost guaranteed if the troops go in) and so on and so forth.
In that sense, they would not necessarily 'Win', because North Korea is small, to use the old adage, they may win the *Geopolitical* Battle, but they will probably Lose the *Geopolitical* War (that is, short term vs long term impacts on Oceanic-US relations)