The OP specified that he didn't want daedric artifacts,,,
The strongest sword in the game is a levelled item (meaning that if you stumbled upon it while you were leve 45 or lower it won't be as good) : Chillrend. It's found during the thieves guild quest. It's an enchanted sword whose base damage is equivalent to a daedric axe.
Speaking of the thieves guild, you get the nightingale blade (unique appearance, glass level stats) and the the nightingale bow which is levelled too: if you get it pas level 46 it's the strongest bow in the game and has a unique appearance.
The best shortblade is Astrid's Blade of Woe: you get it if you join the Dark Brotherhood or if you kill her instead of joining.
There's a orc hammer in the game which is said to be lightweight and swings as fast as a 1h mace. High DPS but no unique appearance. I don't remember where I got it.
Someone mentioned the ghostblade, a unique ancient nord weapon with an unique enchantment: it deals damage ignoring armor: it can even be improved at the forge using... ectoplasm essence!
Another 'unique' short blade is the blade of sacrifice found only at Boethia shrine on cultists: its an ebony dagger with an unique texture.
Chillrend is just a glass sword with a game breaking enchantment. Not very unique imo
The hammer you mentioned is the Longhammer. Forget how you get it as well.
You have to go real deep in the TG line to get Karliah's bow and the sword. Seems odd the TG has so much unique gear associated with it. Three sets of armor and three weapons.
Another unique weapon is Wuuthrad, you get it bear the end of the companions line. It's a battleaxe