So this is pretty much the list I beat the Lich King with. You don't have to copy it one to one - it was actually mostly build from stuff I had flying around anyways.
The main idea is to stall with taunts and then just accumulate bonus attack from Cobalt Scale Banes. Central cards are the scale banes, large taunts (the tar lords are really amazing and if you can get one out and buff it with stormwind champion it pretty much takes care of the Frostmourne phase) and in particular the Obsidian Destroyers. The Lich King doesn't go very wide and very often the 1 taunt per round will keep him completely in check. If you can get two out it can be pretty much game. (props to Tommy J's video https://www.youtube.com/watch?v=5w_fmU7Vufs&t=76s for the idea). Another core card are the armor smiths. It is very possible to get 30+ armor during a game from them (in particular when you combine them with your whirlwind effects) and this is crucial to survive long into his last phase.
Blood razor is an amazing tool to clear his early boards by the way, which is very important to avoid him buffing it with anti magic shell.
If you want to improve the deck a bit, frothing berserker's are very amazing. Brawl can take care of the Frostmourne phase (see Tommy's video again). I also had a doomsayer in my deck, but didn't even use it for the winning run. I had an almost won run as well, where a second doomsayer was stuck in my hand, so two are probably too much (same should hold for brawl). Besides that any very large or efficient taunts are welcome. As said before the Lich King doesn't go super wide, so any minions which produce extra taunts are quite good.
Good luck with the fight - you'll probably need a few tries, but it's satisfying once you did it :)