I’d like to share a number of thoughts after attempting this quest several times, in the hopes it will help others. Sorry for the long post - believe me this was originally much longer 
On a freshly dinged 50 Mage in full heirlooms, I failed miserably. Came back later in full Nazjatar benthics, again failed. Most recently, I tried yesterday on post-squish Level 56 wearing ~iLevel 108 gear, for reference, that compares to pre-SL approx 445 (e.g., The Eternal Palace mythics or Ny’alotha normal drops). I did not deal significantly better hits, so again failed quickly. I usually spec Arcane though also tried Frost. I don’t mean to sound so hopeless, though thus far haven’t been able to complete the quest.
As Cordana and Astyanax observed, this quest appears to scale to your Level and iLevel, and it seems I am no better off with my current gear and higher level than where I was at 50, wearing the same gear most PvE will have starting this.
To be fair, I suspect for some classes (like a Paladin, Monk, Hunter, Warlock, or Priest, with “good” in-combat shields and heals), might fare better. As for me, as an arcane mage without significant heals or long-lasting shields, it seems impossible. If you’re an arcane mage and thought this was easy, please share tips, include your rotation, talents, levels, any special items used to survive.
One option I’ve seen (for mages) was to shimmer outside the room, then kite the boss around the building while pegging away over the next 10-15 minutes, until you eventually wear it down and defeat it (unless they one-shot you, of course). I tried this, though at random, one or more friendly ‘Mechanized Peacekeeper’ bots appear, driving the boss back into the building, resetting the quest.
As an aside, IMHO running a character from 1 to 50 normally without using a boost or race/faction change should merit acquisition of your heritage set, if not after an enjoyable acquisition quest (e.g., Blood Elf). This holds true for all the allied races, where you’re simply presented with the transmog set, done.
If Blizzard ultimately decides to be kind and nerf or change the parameters (and the following Galliwix battle) to make this, well, easier, I would be so grateful.