When you first enter, use your cane to flip the switch on the far right corner to open the door. In the room, there are four flashlight goons patrolling around the slot machines. Silently take them out one-by-one to avoid detection. Collect the clues besides the slot machines, then press on.
There will be two more flashlight goons blocking the path with their lights. Climb up to the ledge and then drop down to quickly subdue them. Collect the clues on the end of the ledge as well as the ones on the floor. Then climb up to the chandeliers to get across to the other area. If you want some extra coins, grab onto the hook attached to the big slot machine and let it lower you down and a big wade of cash will come spewing out for you to collect. Don't forget about the clues hidden in the plants behind the slot machine.
Keep moving forward and you will then come across floor lasers again and there are clues are hanging above them. It is better to avoid the clues and smash the alarm first. As soon as the floors start to blink, start running and jump off as soon as possible to avoid touching the lasers. Take out the guard, collect the clues, then press on.
You will now be approaching the flooded section of the casino. You will have to jump onto a lot of objects to get through and you must avoid the electric floor as well. There are also some small roulette wheels with lasers and floor lasers, which can be avoided with skillful jumping. Continue on and you will be at the end. Collect the last few clues and crack the vault nearby. After that, go ahead and hop onto the big spinning roulette wheel. Jump clockwise to land on the green section. After doing it once, lasers will appear in some sections, so be mindful. Step on the green section three times to unlock the key. Take the key and your work here is done.