To properly end the quest, you have to go back to the starting location, with the blade in your inventory (NOT equipped, or it won't be recognised). The quest description doesn't explain this, which makes it hard to figure out why the quest is still active.

OK, I went back to the original quest location.  First when it "awarded" the Blade to me it now required level 18 when I had been equipping it all along.  Also - and more importantly, the game deleted the Assassin's Blade from my inventory.