Start your exploration of the Western tower but, you need to stop where you will notice the Joker's goons in the distance. This is so because there is an operating defense cannon here. Select the disruptor from the inventory, target the cannon (the screenshot) and sabotage it (by holding down the appropriate key/button). You can now charge the control panel on the right with the shock gloves and unlock further passage this way.
Be prepared to deal with the enemies that rush towards your character right after the metal gate is raised. First, deal with the martial artist and with the remaining bandits, who are not going to pose too much threat, afterwards. After you win the fights, go towards the Southern storage [West Tower - Ventilation Shaft]. Walk along several narrow corridors, jump to reach the small ledge and look up to locate the interactive ledge that Batman can pull up to, using the hook.
After Batman grabs onto the ledge, shimmy to the left and reach the spot where you can pull up. Use the hook once again, reach the balcony's end and extend the rope using the grapple. Walk over the rope, find the control panel and interact with it, which will stop the huge fan. Explore the area above Batman, note the venting shaft and climb up by using the hook on its ledge.
After you exit the shaft, use the claw on the manhole above Batman and go left. Target the ledge above, with a single enemy standing right next to the edge. After you reach the person, throw him off the balcony and, only then, pull up to finish your stroll across the side rooms and reach the door to the West Tower - 25th Floor.
Continue exploring the hotel, deal with the two bandits that you bump into along your way (watch out for one of them who has a knife) and reach the West Tower - Ballroom, which has been turned by the Joker into a devilish amusement park. Jump into the shaft and reach the area where you fight another fight with a group of enemies. It is best to start by throwing the enemies a concussion detonator. Throughout the later phases of the fights, wait for the opportunities to use the shock gloves, especially that you will be able to attack, with them, the armored enemy, without the necessity to stun him with your cape.
Enter the amusement park and note that the countdown has started. You have, exactly, 90 seconds to free the people held hostage by Joker. You need to start by using the batarangs to hit the red buttons behind the hostages.
Use the grapple now, to connect the two poles with the rope. As soon as you do that, pull up to the rope, using the hook and glide towards the small balcony neighboring the weakened fragment of the wall. Blow the fragment up, using the explosive gel, of course.
The last action you need to take within the constraints of the allotted time limit may turn out to be more time-consuming. Therefore, you should make sure that you have enough time left. Select the Cryptographic Sequencer from the inventory, target the control panel and start the hacking mini-game. Unfortunately, Joker has ensured better security measures so, there are many possible combinations and you need to select the correct one (the screenshot).
You may now jump down, because the discharges in the room with the imprisoned disappeared. Turn back and go ahead to find the ladder. Soon, you will reach a place where you need to solve another puzzle. Stop on the pressure plate, turn to face the opening and throw the remote controlled batarang in there. After the batarang makes it through the opening, wait for a moment, turn it by 180 degrees and direct it towards the other opening shown in the screenshot. This is not the end yet, because the controlled batarang needs to hit the fuse box (it is a good idea to decrease its speed here).
Use the hook to reach the newly opened hole directly above Batman. Soon, you will reach the final puzzle in the amusement park and, to start the trial, stop on the pressure plate. The plan is that you hit the balloons with batarangs in the very same order in which they are highlighted. The first sequence is (from the left): 1, 3, 4, 2. The second sequence is: 4, 3, 1, 2. You can now go ahead and jump down after several moments.
Climb up onto the fence enclosing the arena with Joker's men in it. I recommend that you start this battle by using the concussion detonator and eliminating the martial artist as quickly as possible. Then, take to the weaker enemies and use them to charge the shock gloves. You are going to use the gloves to eliminate the armored enemy, which is going to be easy, given the fact that you will not have to stun him with the cape, as a result, and land heavy blows straight away.
After you have won the fight, locate the balcony with an interactive edge and reach it using the hook. Press the button and use the grapple to extend the rope in the spot shown in the above screenshot. You can now walk over the rope and reach the door to the West Tower - 25th Floor. After you get into the room with the armed bandit in it, duck and sneak up to the enemy to perform the quiet takedown and save the hotel employee, that Batman will have a chat with.
Take the corridor towards the Overview Bar and lift any of the two available grates. Walk through the ventilation shaft and perform the takedown on the nearest enemy after you reach its final part. Start here another fight, during which you should concentrate on the enemies with knives and stopping the bandits from acquiring firearms. Just like in the case of the previous major battle, you should leave the enemy with heavy armor for the end, and eliminate him in a much easier way with the shock gloves.