Proceed out the door to take the mech outside of Dark Athena, onto the outer hull.
Outside, you can use the various bumps and blisters on the hull for cover. You'll face another mech at the end of this first trench. Honestly, you can go toe-to-toe with it and come out ahead without a problem.
Move through and take out all the little floating gun turrets. There are a bunch, but they're not very mobile. When that's done, look for a mech repair station to the left, at the other end of the "room."
Two more mechs wait for you in the area ahead. One is above the door at the far end of the room. The other is on your level to the right. Both will toss rockets at you. You'll hear them coming from the alarm that sounds inside the mech.
Fire off rockets of your own at both mechs. They're effective at long range. If you close the gap, you'll have their miniguns to deal with, but yours will be much more effective.
Scrap the two mechs and head through the door. You'll return inside. Follow the hallway, kill the mech that's murdering the prisoners up at the corner, then look for a repair station along the left wall.
Continue through and you'll enter the cell block. Revas makes a hasty retreat with the still-disoriented Johns.
You need to take care of the two mechs in the room. Fall back behind the door frame and wait for them to get close together. When they do, start slinging rockets in their direction. The explosions will cause both mechs to list a little, giving you open shots with your miniguns. You'll also save on rockets because both mechs will be damaged.
When they're both dead, move forward and exit the mech. It's back to sneaking through vents.
Enter the ventilation duct at the far end of the room. On the other side, Silvermann will open cut the grate away for you.
You'll have to fight Jaylor next. He won't come out until you put away your weapons. After you do that and he appears, however, feel free to kill him with whatever you're carrying -- unless you want an achievement.
Achievement Alert! Beat Jaylor with only your fists to earn the Honest Fighting Achievement.
The best way to deal with Jaylor is to be extremely cheap. You can try to bob and weave, stepping away from his punches, but this affords him too much opportunity to counter. It also gives him the chance to hit you and put together combinations. Neither of which is fun.
Instead, stand in place, continually facing Jaylor, and keep your guard up with Left Trigger. You'll block every one of Jaylor's attacks. As soon as you do, follow up with a quick tap of Right Trigger to pummel him. Repeat until he dies.
Jaylor can take a lot of punishment, but if you use the above strategy, he probably will only barely hurt you. When you're done, take the door out of the room with the green light, toward the comm station with the big fan. You'll get a cutscene along the way.
Proceed toward the comm station. You'll be ambushed by one of the other prisoners along the way. Take her out quickly and then go to the console.
Dacher tells you to head to the elevator to the hangar. Backtrack from the comm room back to the control room outside Cell Block 12B. Head toward where Silvermann's body is and look for the keypad along the way. Use it and you'll open the elevator.