This is directly taken from my walkthrough on this site:
-Now, from the big gate, head straight south. There are two sets of staircases, one to the right and one to the left. Head to the right, and use Trumpkin's grappling hook to pull the door down. Head up this staircase, and watch the extremely short cutscene.
-Lower the Drawbridge: Pull the lever right below you to lower some stairs. Go up these stairs and kill the group of telemarines that attack you. Keep heading along that pathway until a cutscene starts, showing you our final destination. Continue on, first having Trumpkin use his grapple hook to extend a rock bridge, and then Reepicheep climbing the gated door to open it. Continue on, and go through the small hole left of the gate you need access to. This takes you to a lower balcony, where you will again use Trumpkin's grapple to pull a rock bridge. Cross it, hit the pressure pad, and enter the hole right next to the pressure pad. This takes you to a new chest, costing you 5 keys (Chest 3 of 11). Go back through this hole, and the one where you first got to this lowered area. Use Reepicheep to jump up the rocks in the wall, following the button prompts. Head through the newly opened gated. As soon as you enter, kill all of the guards, and open the chest for 2 keys (Chest 4 of 11).
-Continuing on, stand on the circular platform at the base of the raised platform with the levers. Your AI partner will then raise you up, so hop off and get ready. These levers need to be pulled in a particular order, so from left to right, pull the first, second, first one again, and then the third one. If done correctly, a cutscene of some gates being opened will play. Exit this room, and retrace your steps back across the walkway (killing all the guards along the way). While heading back, enter the second hole in the wall (the first is where you just came out of that room). Use Trumpkin to make a bridge, and enter the room on the other side. Kill the guards, and enter the walkway south of the room. Here, you will find a chest worth 6 keys (Chest 5 of 11). Re-enter the room, find the gear on the left hand of the room (up some stairs), and put it where it needs to go on the floor. Right beside some stairs in the northwest corner of the room, destroy some barrels to find and step on a pressure pad. Head directly right, and in an alcove, there is another pressure pad: step on it. Use Reepicheep to jump the ledges in the wall to reach the gear, and knock it down. That was a close call. Using Trumpkin, put the gear (found next to entry door) in it's location on the floor. Now, start pushing the crank for a mini cutscene. Kill the guards, and continue pushing the crank until it's mission over