Purging the Chambers of Bash'ir

This quest is repeatable (not noted at the time of this comment).

You know, as nice as it would be to be "keyed" for Yor in heroic mana-tombs, I cringe every time I get one of these keys made. You get 250 rep for every cage you open with a key, or combine 5 of them together, complete this quest and get 500 rep. 1250 > 500.

Maybe I should wait until I'm exalted when rep doesn't matter so much.

Here is my experience summarized on how to get this item. I just did it on one of my characters to prepare for Timewalking in patch 6.2.

Before you start you might consider purchasing the required materials that will be required further into the quests. It can help reduce farm and speed up completion. At some point you'd need to turn in 10x[item=29460], but in addition you'd need one extra to continue the quest chain. So if I were you, buy at least 10 of these from the Auction House. Though this is not required, just saves you perhaps an hour of farm in total (if unlucky like me.)

Anyway start by doing these quests in the following order:
1. Seek Out Ameer (requires Honored)
2. A Mission of Mercy
3. Ethereum Secrets (repeatable, do once)
4. A Thousand Worlds (requires Revered)
5. Stasis Chambers of Bash'ir
6. Purging the Chambers of Bash'ir (repeatable, do once)
7. The Mark of the Nexus-King
8. Stasis Chambers of the Mana-Tombs
9. Nexus-Prince Shaffar's Personal Chamber (repeatable, do once)
10. The Eye of Haramad (requires Exalted)

One tip to greatly increase the speed of the farm is to actually enter Heroic [zone=3792] because you can farm the last boss indefinitely. He will first spawn [npc=18431] and these mobs will spawn [npc=18430] and these give 10 rep in Normal, and 21 rep in Heroic. If you have 4 hours to kill spamming one macro, basically create this:
/cleartarget
/targetexact Ethereal Apprentice
/use [harm,nodead] ABILITYTHATCANBESPAMMEDTOKILLSTUFF
/stopattack
/tar player
/use Battle Standard of Coordination
I've included the [item=64399] to speed up the process. Also helps you not forget to spawn it once it is off cooldown and the old one despawns.

Once Exalted you can go to [npc=22919] and complete the final quest The Eye of Haramad (only requires Exalted, nothing else) for [item=32092] and you can spawn [npc=22930] each time you visit the instance - no need to farm keys for one time use consumables.