What The Dragons Know

"There's no mistaking it - this hilt comes from a powerful, ancient weapon, but which one? I do not recognize these particular markings, but we haven't hit a dead end yet."

Really? What about that sword missing a hilt THIRTY FEET IN FRONT OF YOU?

Here's the entire quest chain, with simple descriptions and quest links for the Horde versions.

1. 
   
2. The Battered Hilt
   Bring the Battered Hilt to Myralion Sunblaze at Quel'Delar's rest, south of the Argent Tournament Grounds.
3. 
   
   2.
4. What The Dragons Know
   Speak to Krasus at the top of Wyrmrest Temple.
5. 
   
   3.
6. The Silver Covenant Scheme (Alliance version is called "The Sunreavers' Plan")
   Obtain the Orders by killing an NPC in the Dalaran Sewers. Make sure you kill the right one.
7. 
   
   4.
8. A Suitable Disguise
   Obtain a Borrowed Tabard from Shandy Glossgleam; He is upstairs in the Threads of Fate: if you are standing in front of the stairs with the Violet Citadel to your back, it is to the right (to the left is the toy store).
9. 
   
   5.
10. An Audience With The Arcanist
    Retrieve the Ancient Dragonforged Blades tome from Arcanist Tybalin. Use the disguise and go in the Alliance section of Dalaran.
11. 
    
    6.
12. Return To Myralion Sunblaze
    Bring the Ancient Dragonforged Blades to Myralion Sunblaze at Quel'Delar's Rest.
13. 
    
    7.
14. Reforging The Sword
    Obtain 5 Infused Saronite Bars and the Forgemaster's Hammer and use them to make the Reforged Quel'Delar. (AKA: Run the Pit of Saron. The infused bars are near the entrance; don't forget you have to kill the boss, too.)(It appears this can be done on Normal Difficulty.)
15. 
    
    8.
16. Tempering The Blade
    Temper the Reforged Quel'Delar in the Crucible of Souls. (AKA: kill the final boss of Forge of Souls; use the quest item nearby.)(It appears this can be done on Normal Difficulty.)
17. 
    
    9.
18. The Halls Of Reflection
    Bring your Tempered Quel'Delar to Sword's Rest inside the Halls of Reflection. (AKA, obviously, run Halls of Reflection.)(It appears this can be done on Normal Difficulty.)
19. 
    
    10.
20. Journey To The Sunwell
    Speak with Halduron Brightwing at the entrance to the Sunwell Plateau on the Isle of Quel'Danas.
21. 
    
    11.
22. Thalorien Dawnseeker
    Visit Thalorien Dawnseeker's Remains and obtain his blessing to wield Quel'Delar. (Go around behind SWP to the Dead Scar (southern end of Isle of QD zone, not in the SWP raid instance) and kill a bunch of mobs.)
23. 
    
    12.
24. The Purification of Quel'Delar
    Immerse your Tempered Quel'Delar in the Sunwell. Speak to a Warden of the Sunwell when you are ready to enter the Sunwell Plateau. (Possibly different versions of this for sword/mace?)
25. 
    
    13.
26. A Victory For The [Sunreavers/Silver Covenant] - (sword or mace)
    Bring the Restored Quel'Delar to the NPC at the Violet Citadel in Dalaran.

A lot of this info came from comments people left on the various quest pages. Thanks to all those who posted useful tips!

The in-game quest feature points to the wrong area.

You want http://www.wowhead.com/?npc=36670 who's in the horde only Inn.