When you have the superiority the US sports, it's not hard to win a war. It becomes a question of what you're willing to do to win it.
The US is one of the only countries in history to have to deal with that question. Defeat Iran? Easy: nukes, or a never-ending bombing campaign that leaves nothing left standing. Throughout most of history, the problem with defeating an enemy wasn't a question of: well, but can we do it without killing too many people.
North Korea's army is worthless. They're extraordinarily backwards, poorly trained, and with mediocre equipment. The problem would be the guerrilla warfare part, much like Vietnam.
Iraq had one of the largest armies as well, and the world watched in shock as the US trivially dismantled it. The US would dismantle the standing North Korean military very quickly, it would be the next stage of the fight that would be terrible.
If the US had a reason to really want to topple North Korea, the easy thing to do would be to make a deal with China: they get Taiwan tomorrow morning, and they allow North Korea to collapse, don't get involved if the US fights a war with North Korea, etc. China would make that deal in a heartbeat.
The biggest problem in fighting a war with North Korea, as everyone knows, is the damage that would be done to South Korea.