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If you do this on multiple toons like I did you might now realize that Wrathion won't be there to talk to in his whelp form unless you complete the unlock the forge objective part of the quest by doing the first couple of scenarios you see in the front of your factions part of the island. Took me forever to figure it out.

Edit: This quest, as with all of Wrathion's other quests, was removed from the game with Patch 6.0.2.

For an overview of all Stages, you may watch this video on YouTube. (Video length is 26:40, contains a few strong words.)

Stage 5 of the solo scenario is a real pain in the rear because of its RNG. You must protect the Celestial Blacksmith with the aid of a Celestial Defender. Naturally - and despite his earlier words about dwarves and button-pushing - Wrathion pulls a Brann Bronzebeard and is busy fiddling with the machinery; he won't be of any help to you.

If you are able, eat a useful stat +275 food and quaff a +1000 flask.

Know Your Class (And Spec) really applies here. The event requires that you eek out every point of damage you can while tightly timing your survival cool-downs. The event will fail if the Celestial Blacksmith dies or if you die, and you will have to try again. Useful tips: Target and right-click the blacksmith's portrait and select Set Focus. Also, right-click its portrait and choose Target Marker Icon, then select one of the icons (like Star).

The Celestial Defender helps you by tanking adds and occasionally creating green beams that heal you. Use these beams wisely. When the Celestial Defender is low on health, it will go into stasis to regenerate its health. This causes it to lose all aggro on any mobs attacking it, so you need to be ready to pick up aggro, kite (and hopefully AoE DPS) until the defender is back online.

Thundering Anvils will overcharge over time. Discharging them will kill all Sha in the room. You have a limited number of Thundering Anvils that overcharge, so discharge them wisely. (I saved them for the Sha Beasts.) An overcharged anvil also means that the Blacksmith will move to another anvil, so keep up.

Eventually, the Sha Fiends and Sha Beasts will stop spawning. (The energy indicator bar at the bottom of your screen will be full.) But you aren't out of the clear just yet! A Sha Amalgamation spawns for the final stage (Stage 6).

• DO NOT USE THE ANVILS FOR DPS BOOST. You will need the anvils to interrupt its  Insanity. (Pro-tip: If you can, use the overcharged anvil furthest from your position first.) As you have a limited number of overcharged anvils to work with, the Sha Amalgamation essentially has an enrage timer.
• Avoid getting hit by Shadow Crash.
• Run around until Meteor Storm is over.
• Kill the summoned Debilitating Sha! It stuns the Celestial Defender.
• Be ready to run when  Dark Binding is cast. The mob is immune to stuns and snares, and moves slightly faster than you can run.
• Pray to the RNG gods because this fight has a lot of RNG. Like healing beams that spawn in unreachable places. /facepalm

If you die during the final stage, you only have to repeat the final stage. IMPORTANT: Resurrect as close as is possible to the door if you don't want to be immediately pulled back into combat.

For an alternative strategy for Stage 6 involving speed, read Turpentine's comment on Wowhead.

Wrathion will have the lance ready at approximately 20%HP (9M) left on the sha. Run over to it, click it, target the sha, and click the Special Action button at the bottom-middle of your screen. Do not move because the lance is a channeled ability.

Congratulations!

Just took my healer back in and kept my heals on the Defender the entire time keeping him topped off in addition to bringing him back to 100% when he is down to 50%.  With extreme healing on the Defender I believe was my success.  Sha went down to 20% and the lance appeard, I clicked it, and the button appeared on my action bar.  While I was channeling it, Sha started his 8th Insantity but I beat him to the punch and he is now dead!!

I am a druid healer who has tried this about 80 times and failed because Sha never gets down to 20% and I don't have enough anvils to interrupt the 8 Insanities.  I am constantly healing the defender so his health is full or near close to it to get better dps out of him.  In balance form, I dps with very little heals on defender and still can't get him to 20% and I run out of anvils.  My gear breaks about the 7th death, so I have to start all over.  I don't have any repair abilities.  What am I missing?  Is there a way to reduce the number of Insanities?

Good God, just how are we to Solo this if we can't DPS enough?

Well, you can try to improve your average item level. You can also have friends cast beneficial abilities such as Blessing of Kings, Fortitude, etc. on you prior to entering the Thunder Forge solo instance - you can't be grouped in a party to do this, though.  You can also try changing around talents and glyphs.  Good luck!