DCUO Boss - Supergirl
When you first confront Supergirl, you aren't meant to actually take her on
at this point. Ignore her and whenever the scientist encases a patient, get to
it before Supergirl can release her. There is a spot in the middle of the room
where you will need to toss the encased patients. Every now and again, Supergirl
will trap the scientist, and at this point you need to free him quickly so he
can keep zapping sick people. When you've gathered all of the patients, talk to
the scientist to progress.
To disable to security door, grab one of the orange/brown barrels and chuck it into the designated location. From there, go back to the newly opened area and open the doors using the big contraption on the west side of the room.
In the next room, again completely ignore Supergirl. She'll attempt to trap you so break the trap and keep destroying the computers. It's not time to fight Supergirl just yet!
Finally, after disabling another door and dealing with insane patients, in the final room you get to take out Supergirl. Before you can actually fight her, you need to find the Kryptonite. I found it in a breakable barrel next to the middle console. Keep her close to the Kryptonite and slowly knock down her health. Avoid her laser and ice trap and be sure to have Soder Cola Classic (you've outgrown just regular Soder Cola at this point) on hand to keep your own health up during the fight.
If you picked up the mission at the beginning of the zone, you've checked out the terminals throughout the zone and need to talk to that guy before leaving.