Ok, so I worked on my one handed weapon skill/perks for this sword and now I read in the Wiki that it also benefits from the two handed weapon skill but I just purchased 5 levels from Vilkis and there was no change in it's damage. Can someone who knows please explain how this weapon works?

I don't think two handed would benefit oen handed in anyway. I would guess its why they are two different categories. I stick to one handed, and never put anything into two handed. I also use enchanted increased one handed damage as well, to help me. I'm not quite at level 80 for one handed, but I am doing about 230 damage for one handed with daedric sword/mace. I have 4 items of apperal on me that adds 40% one handed damage. I would reload a save if I were you, to save the money you used on vilkas, and use it on another guy in the companions, who does one handed training. I can't think of his name, but he almost has a mohawk/pony tail like hair cut. The name should start with an A too

aithis

Im talking about the Ebony Blade which is a two handed Daedra weapon that goes by your one handed perks but the Wiki is saying that it is also affected by your two handed weapon skill so Im confused. I did save it before I bought the 5 levels from Vilkis though.

Just make a one handed Daedric Sword. Problem solved.

The Ebony Blade - the one from the Daedric quest is a two handed based weapon so leveling the skill is possible through the use of this weapon. Its damage is based off your one handed skill as well as one handed weapon enchants , potions , Etc so no the two handed skill doesn't affect damage of this weapon , only one handed does.

Hope this helps