I'm currently doing the quest to reforge Trueflame in Tribunal. I am trying to get the third piece of the blade from Torasa Aram, but she won't give it to me. I have donated the Dagger of Symacchus and the Vampiric Ring, but to no avail. The UESP Wiki even says that I only need to donate two items. What's going on here?

It's possible that you actually sold the 2 artifacts to the Museum, as apposed to donating them. I made this mistake on my first Tribunal playthrough.

Normally (when not on this particular quest), you have 2 options at the museum: Sell an item, or just keep it. When this quest is active, a third option is enable: Donate the item. You won't get any money for donating, so make sure to donate artifacts that you don't like.

Do you remember if you were paid for the items? If you were, this is where your mistake is.

I chose to donate them.

It is known that the count on donated items sometimes messes up - try donating a third item - that usually solves it.

I tried donating my Ebony Mail (which I wear) as well, which sums up to three items. Didn't work. I guess I'll go all the way back to Caius' house where I stashed the other items then and donate as many as I can until she gives the the darn thing.

I just tried donating five items (Dagger of Symmachus, Staff of Hasedoki, Staff of Magnus, Ebony Mail, Vampiric Ring) yet I STILL can't get her to give me the piece. Am I clicking the wrong dialog options in the conversation or something? This is making me really mad.

Are you on PC? If so, try http://www.uesp.net/wiki/Tribunal:The_Blade_of_Nerevar#Artifact_Museum_Cheat:

could be a bug for some. it took three for me. i thought it was because one of the items i donated was cheap.