North Korean minister of defense executed

[Napoléon]

Honestly, at this point, it seems like the only reason the West doesn't destroy this guy (besides the fear of reprisal from China) is because, quite frankly, it's only a matter of time before the whole nation self-destructs.

There's a bunch of reasons for this:
* As seen in the OP, Kim Jong-Un is hell-bent on destroying any possible rivals, almost certainly leading to one hell of a succession crisis when he died.
* The nation has hardly any natural resources and the vast majority of the people in the nation are starving, thanks to Kim dynasty's policies.
* For all the sabre-rattling they engage in, North Korea's not likely to be able to actually do much power; if so much as one shell hits Seoul (and I'd be very surprised if their nukes can get much farther than South Korea), the armistice that put the Korean War on perpetual hold is over, and, given that nothing short of a Zerg Rush could lead to a North Korean victory in the face of the opposition of, well, most of the world, North Korea's existence would also be over.

I don't think it's because the west thinks it's a matter of time before the nation self-destructs. Not in the slightest.

I think it's more because nobody actually cares about North Korea enough, or are concerned by them enough, to 1: consider them a threat, and 2: consider them worth 'sorting out'. 

The fact they have hardly any resources is the main reason the west isn't bothered about getting involved IMO. It's not viable for anyone at this point to waste their own money, resources and men on a country like NK. There's plenty of other countries around the world, namely in Africa, where human rights travesties happen all the time, but you don't hear as much about them because those countries aren't as brazen and wacky as what NK is. Only reason there's even remote interest from the west is because it sells newspapers and creates funny headlines. 

Add to that, like you say, they don't actually have much power. Their threats are consistently considered a joke by the west. I don't think any major western nation is really concerned about North Korea. There's more important things on the agenda than a relatively small crackpot nation spouting threats constantly.

I also really don't think the nation is in trouble of self destructing. Comparing it to other nations, or empires, in this day and age, doesn't seem too relevant. The Kim family has had a pretty firm grasp on NK, no doubt even if this current nutjob gets ousted someone else will take his place. The whole crazy persona that NK has won't change because of that and the top tier of people with power don't look like crumbling any time soon without outside input. 

Quite simply I think the reason the west doesn't get involved with NK is because it's just plain more trouble than it's worth, not because the nation itself is in danger of crumbling any time soon.