Looks like it's a glitch around how the Harvest of souls HP works. The HP makes him immune regardless; there's a trigger tied to frostmourne that says that if all trapped souls are dead, destroy this and get the third HP; as frostmourne was stolen, you get the phase 3 HP and LK does nothing, stuck in P2. Sounds like he's destroyable with stuff like mecha'thun, though.
ETA - this is judging from a wiki. Just don't use stickyfingers unless you're going to mecha'thun, I guess.
It's be cause they don't revisit old adventures. They added armor to bosses because people could use Alexstrasza and win easily, now they printed Platebreaker and forgot to ban it in adventures. That's how I beat lich king with almost every class. The weapon is immune to weapon destruction and is not supposed to be stolen. You need to focus on his minions, not on the weapon.