Unbreakable is the name of the achievement. You have to kill the Big Sister.
In addition to Doc Buddha's solution about stocking up on First Aid Kits (which you'll need on Hard Mode because two hits after upgrading health and you're left with a sliver of health) and directions on when you pop this achievement after getting the Incinerate Plasmid, don't forget stock up on Eve Hypos.
Big Sister has 3 attacks that I've noticed (feel free to comment if you've seen any more and I'll put them in the solution) :
1. Melee - she rushes you and slashes you for a lot of damage. This is the attack you want to avoid.
2. Incinerate - A Plasmid attack just like yours but easily avoided if you have some distance on her.
3. Telekinesis - Another Plasmid like yours only she doesn't have to have anything near here to pick up.
I reloaded my save prior to fighting her a couple of times and the easiest way I found to defeat her was Electro Bolt to Stun her and then pop her with Heavy Rivets using the Rivet Gun. Hopefully you've saved some Heavy Rivets, but regular rivets will work. The Rivet Gun in general seems to do more damage to armored enemies than your Machine Gun. If you've upgraded the Rivet Gun's damage, even better (I upgraded the Machine Gun).
There is a puddle of fuel on the floor near the bench in the middle. Be sure to use Incinerate on her while she's in the puddle of fuel. Stunning her with Electro Bolt while she's in the puddle of fuel is a nice tactic to give yourself time to Incinerate her.
Other than that, just keep your distance. Run backwards while circling the obstacle in the middle of the area where you're fighting and always face her. Stun her with Electro Bolt and fire off Heavy Rivet or Rivet headshots. The most important thing of all:
DON'T FORGET TO LOOT HER CORPSE. She carries ADAM on her. If you were short some ADAM and you wanted another Plasmid or Tonic when you visited the Gatherer's Garden the first time, run back and get it (I went back and got Armor Shell because I was taking a beating throughout the entire level hehe).
Good luck and hope this helps. If people have too much trouble with this, just let me know via PM or comments and I'll make a video (which I may do anyway).
EDIT: Guh... I thought I added points and tips from peeps commenting on the solution, but I guess I didn't. Sorry, guys! lol.
Here are some more tips:
"When I fought her she used her telekinesis to pick up rubble. It would spin around her head for a second then shoot at me. I'm playing on Normal and did this on my first try." -blomphil2
"Good points but I like using the drill rush attack, follow up with melee, then electro bolt so you can back up and not get hit running away. Then again I might be a little biased since I upgraded the drill fully, researched some stuff to up the damage, and then use tonics to improve it. lol But it is still pretty powerful." AuraOfRage