We know they are capable of detonating a bomb which has the power of a nuclear weapon. We know they have the fuel needed for a nuclear weapon. We have not yet seen direct evidence of a nuclear test, but all the signs are there that they do have them, and they do work.
They are relatively low yield compared to advanced nuclear technology held by the US, china, and others.
We suspect, but do not know, that their technology isn't advanced enough to fit, size and weight, into one of their ballistic missiles, which we do believe are now capable of just barely reaching LA.
So while it is possible that they have a nuclear warhead that fits onto a ballistic missile that hits LA, it is very, very unlikely.
However they probably could launch their current nuclear warheads and hit Japan, Guam, and South Korea.
They could not follow it up with much more than conventional firepower though. They have enough material to build a handful of warheads, but again these are very low yield warheads. Probably comparable to our bunker buster nukes, certainly nowhere near the bombs we dropped on Japan. So they could pick-mark South Korea, and kill millions depending on where they aim, but they couldn't decimate South Korea with nuclear weapons, nevermind annihilate them the way we could.
Their conventional firepower is sufficient to do significant damage to South Korea.
And their nuclear weapons will only get better.
I think they honestly believe that now that they have nuclear weapons they belong at the big boy table, despite the fact that the weapons pose little to no risk since our missile intercept systems currently surrounding them are sufficient to stop a large percentage of any missiles they send out, nevermind the slow ballistic missiles they have.