Weapon type Attacks per second Relative damage
Dagger 1.2 -3
Sword 1 +0
War Axe 0.9 +1
Mace 0.8 +2
Your primary hand seems to attack about 15% faster. The break-even point in DPS between swords, axes, and maces occurs at about 10 sword damage. This is equal to a unimproved Dwarven weapon or a Superior Iron weapon. At or above this level swords will do damage at a faster rate, all other things being equal. This is not, however, counting the weapon specific perks. A mace, with all relevant perks, will hit a heavily armored foe for significantly more than a sword.
Swords having a 20% faster swing speed is a pretty big deal. A Daedric sword has a base damage of 14, where a similar Daedric mace hits for 16. Even in that state, completely unimproved, the sword is superior (sword DPS is 14, while mace DPS is 12.8). Factor in that improvements will decrease the relative damage difference, and it's not even close.
Unless power attack swing times are standardized for all One-Handers, I don't see how maces can ever be make up the difference, especially given the significantly higher weight and stamina drain of maces.