In this chapter, Batman infiltrates the Joker's Arkham City Headquarters, saves a doctor, beats up a hammer-wielding freak and...well, let's not ruin the ending just yet!
You should start the level on a tightrope. Glide over to the area in front of you and turn right. Here's where you need to learn how to slide: simply hold RT/R2 while running to slide underneath the leaking pipe. Vault over two more obstacles before you reach a balcony that's hot to the touch. Get out your Batclaw and use it to open up the circular vent in front of you, which should cool the railing after a few seconds. Climb onto it and drop down onto the floor below.
Turn around and run towards the lava. When prompted, hang along the ledge and shimmy left. Pull yourself up and walk along the railing, then jump across to the railing on the right and continue forward until you reach a dead end.
The floor below you can be destroyed with Explosive Gel, so spray some on said floor and use RB/R1 to detonate it.
Once you've dropped down to the lower level, crouch down and enter the next small area. There are some more pipes spewing steam. Get out your Batarang and hit the switch on the right to turn off the steam. Follow the path underneath a group of Joker's gang discussing whether their boss is really sick or not. Keep going until you reach two more pipes. This time, they're not emitting steam, but flame. There's no way to turn them off, but they do periodically shut off on their own, so time your movements to run past them when they're not spewing burning hot death. Take your first right after the second flame pipe, and eventually the perspective will change, showing the size of the gang for the first time.
When the game switches to third-person again, head right. The pipe on your right is steaming again, and you need to turn it off. The game wants you to use Detective Mode to track the pipe to its source, but there's no need for that when you have this guide! Simply turn left and crouch down again for a bit, then run forward until you hit the wall. On your right are three switches that need to be activated at roughly the same time. This can be achieved by rapidly tapping LT/L2 to unleash a flurry of Batarangs that hit all three switches. Now backtrack towards the now-dormant pipe to enter the grates directly underneath the central Loading Bay.
Here's the first really big fight in the Mill. Use a takedown on the one inmate standing next to the grates, then fight the six remaining inmates head-on. Two of them will occasionally throw crates at you. While it's possible to counter them, it won't count as a Critical Strike if you throw them back, so my suggestion would be to dodge them if you're trying to get a lot of Experience Points out of this fight. Regardless, this is another great place to practice the timing for your Critical Strikes. I find the best timing window is immediately after an attack has landed. Also, don't forget that the direction the analog stick is pointing will determine who you fight next. It's not the best idea to try and strike an enemy who's already in the middle of a counter-attack.
Once the group is taken care of, you're treated to a short cutscene with Harley Quinn. After that's done, head up the stairs to your left to trigger a radio conversation with Alfred, then go through the door.
There are two Riddler trophies on the left here, one for you and one for Catwoman, but you can't get them yet until you acquire an upgrade to your Cryptographic Sequencer. If you want, hold LB/L1 while you're facing the trophies to tag them on your map. Head through the next door.
Hit the switch on your immediate left to open up a hatch. Go through it, turn right and grapple up to the next area. Immediately Grapnel up to the nearest gargoyle.
This stealth section is very easy; there are only three enemies, all of whom can be taken out silently. First, wait until the inmate directly below has moved to the left before dropping down and doing a silent takedown. Move down the stairs to sneak up on the two remaining guards, and get them both in a Double Takedown.
Next you want to enter the room directly behind the two inmates you just clobbered. The door's locked, so dive through the window instead, and enter the door on your right.
This next area needs to be done stealthily, otherwise the doctor being held hostage will die. As soon as you enter, wait until the guard on the left has walked up to his buddy before Double Takedown-ing them, then zip up to the nearest gargoyle. There should be a guard right underneath the gargoyle on your left; get over there and use an Inverted Takedown (Y/Triangle) which will attract the attention of one of the guards on the main floor.
Move to another gargoyle, wait until the guard is distracted with the Inverted guard you caught earlier, and take him out. One of the two guards holding the doctor will go investigate, and Inverted Takedown him too. Now you only have the one guard holding the doctor to deal with. Move to the gargoyle directly across from the hostage and use your Glide Kick to incapacitate the guard, before using a Ground Takedown to finish him off. Free the doctor to start another cutscene where you acquire the Remote Electrical Charge.
Among other uses, the Remote Electrical Charge and destroy the statues of Harley Quinn scattered around the Steel Mill, which you'll need to destroy if you want to certain Riddler Challenge. The REC can also open certain doors and stun enemies, causing them to attack their allies.
After rescuing the doctor, you should have levelled up. Put the upgrade into whatever you want. In my video, I chose to upgrade my Combat Armour, but really it's up to you.
Zip up to the nearest gargoyle. There should be a shutter directly across from you, slightly to the left. It's closed, but you can use the REC on the battery panel above it to force it open, then walk through the door behind it.
Assembly Line (again)
Zip upwards and enter the air vent. You can now test out your REC in combat, which can be quick-fired with LT+B/L1+Circle. It's useful on tougher enemies, but slow-firing and clumsy in close-quarters. After dealing with them, you'll notice the inmates have welded the door shut, so turn around and go through the now-broken window and across to the other door. You should hear Hugo Strange announcing the countdown for Protocol Ten. Head through both doors towards the Loading Bay.
Loading Bay (again)
Run past the still out-of-reach Riddler trophies and open the next door to the area where you fought the large group of inmates earlier. You'll notice the large battery panel in the centre. Use the REC to turn it on. The charge will move a large crane, which you need to manipulate to smash through the Joker's protection on the top floor. After this simple physics puzzle, attempt to zip up towards the Joker's hideout, where you'll be kicked back down to the ground by the first boss, Mr. Hammer.
Mr. Hammer can be a formidable foe, unless you know what you're doing. The only way to deal damage do Mr. Hammer is through a Beat Down, which is normally inadvisable because it leaves you vulnerable to the regular inmates who are attacking you. The best way to deal with it is attack Mr. Hammer first, and Evade (Left stick + A A/X X) whenever you see an attack from Mr. Hammer or an inmate coming at you. You should easily be able to get a 50x combo from Mr. Hammer, and then it's a simple matter of dealing with the rest of the inmates. Once everyone's unconscious, you'll hear Harley Quinn crying, so zip up to where you were trying to go in the first place, watch a cutscene and the level ends.