The Quest, "An Axe to Find" is unable to be completed on my game.

After entering the East Empire mine and getting all the way to the end and finding stahlrim, I went to search for an ancient Nord Pickaxe.

The quest that should give me said axe, "An Axe to Find", leads me to the same NPC involved in the mine quest.

The marker shows that I can find him in his house, but the door remains locked at all times.

I've picked the lock and the quest marker just leads to the middle of the room, and the NPC is nowhere to be seen.

The NPC and his wife are also missing from the mine (where I originally encountered them).

Any thoughts on how to correct this/where to find the NPC?

The couple you are refering to are also part of the quest concerning Final Descent. Have you tried clearing the cache and re-loading?

I never cleared the cache, but after completing the Final Descent quest, and giving up on An Axe to Find, I decided to go dump my loot in my house at the mages' college in Skyrim. When I returned, the guy and his wife were both in his house, and I was able to succesfully complete the quest.