I'm telling you, use nothing but fists, or you can pick up oblivion GOTY edition which was the game before skyrim, it doesn't look as good but it has a bit more skills and stuff you can train up, one of which is a dedicated hand too hand skill set. ( granted
being an archer in oblivion sucks)
My advice would actually be go and get some more games! Skyrim is great, but variety is the spice of life and there's plenty of awesome titles out there. Unless you actually already have other games, in which case ignore this entire statement.
If you only want to use a straight-up class then, well, yeah, you've pretty much covered the options. You could try being a necromancer who only fights with a knife should things get hairy. As US says, try fighting barehanded for a real challenge.
Go down the Ork path and become a berserker wielding massive weapons but little armor. Become a Battlemag or a spellsword! or a Rogue that goes for the close kills.
I love the melee gameplay in Skyrim so my favorite character is an Imperial that specializes in swords and shields. If anyone is going to go hand use the Khajit character. Their claws give bonus damage.
you could also do what someone on youtube did and make a "monk" character, who never killed anything the entire game, he use healing spells to not die, ran from fights, ect. see how much you could do with not killing a soul?