First the only connection between room 51 and 53 is that they are connected by a hallway, one area does not affect the other, just that you have to pass area 51 first to get to 53. Note: I will use the compass direction as stated in the map Now area 53 is like this, when the party arrive they will see a room with a wheel on the wall and a lever (west wall, drawing that looks like a triangle with a stick in the middle), on the north-east side of the room is the exit trough the stairs, but is block by some gate. If one character goes where the wheel is and turned the gate on the other end of the room start going up, at the same time a pit in front of the wheel will start open. Any character in the area of the trap will fall 30 feet (3d6 damage), if the character on the wheel turn the wheel slowly with a perception check the other characters can figure it out that the floor is opening, and with an acrobatic check they can avoid the pit. The character operating the wheel has a 1 foot ledge after the pit is open so he will stand there. once the gate is completely open and the pit the other character can get out through the stairs but the character on the wheel is trap. If he pull the lever that blocks the wheel in position the pit and the gate will be open, but as he pull the lever a second gate comes from the celling and close the exit. A character can attempt to lift the gate but they need a DC 20 Athletics check. If they are successful, the character on the wheel can cross over the pit using the 6 foot bridge that appears once you activate the lever to lock the wheel. this 6 foot bridge is difficult terrain so his movement cost double (10 feet of movement for each 5 ft.). remember that the area has a check to cross the bridge. The other way of opening the second door the falls is to find the key on area 48. let me know if this clarify a bit. I'm running the same adventure and my players are about to get there. I hope this help
