Open Deep Dish Dungeon's Three Lever Prison Door
Repair the three prison lever mechanisms with Iron Rods, find the Chains on the lower route, and open the large counterweight door.
The large prison door opens after you repair all three lever mechanisms and pull them. Bring three Iron Rods for the broken levers, then fit Chains to the lone mechanism beside one of them. If you are staring at the counterweights and nothing is moving, the missing Chains are usually the problem.
This is the later prison puzzle with three levers. It is not the first locked prison entrance, which uses the Prison Key. That distinction matters because the dungeon is very good at putting two different problems in rooms that look equally miserable.
Count the broken levers before using your Iron Rods
Check the whole room first. The large door is tied to three lever mechanisms, and the damaged ones each need an Iron Rod. One rod can be found in the right-hand cell after you enter the prison, while more are available along the lower route.
Do not spend every rod on unrelated repairs before you finish this room. Carry three if you can. That saves you from fixing two mechanisms, discovering the third one is still dead, and hiking back through the prison for another piece of metal.
Repair the lone mechanism with Chains
One lever sits apart from the others and needs more than an Iron Rod. Fit the Chains to the mechanism beside it, then repair any remaining broken lever parts with rods.
If the interaction prompt still says the mechanism is incomplete, make sure you are aiming at the machinery next to the lever rather than the handle itself. The lever cannot lift its counterweight until that repair is finished.

Take the lower prison route to find the Chains
Head back toward the stairs where you first entered the prison. Use the nearby stairs to descend into the lower section, then open the route below. This area contains another Iron Rod and a Pickaxe, so clear enough inventory space before you start picking through it.
From the lower route, take the rope on the left back up. At the top, look for a small tunnel leading to a chest. The chest contains Nails and the Chains needed for the prison mechanism.
The easy mistake is searching the lever room for a loose chain. It is not sitting beside the door. You have to leave the room, take the lower path, climb the left-hand rope and check that upper tunnel.

Return to the door and pull all three levers
Bring the Chains back to the isolated mechanism. Repair it, finish the other lever repairs with Iron Rods, then pull all three levers. Once every counterweight is working, the large prison door opens and you can continue toward the elevator and the next set of cells.
If the door stays shut, run the room in a simple order: check each lever for a repair prompt, confirm the Chains are installed on the lone mechanism, then pull every lever again. Most failed attempts come from treating the three handles as the whole puzzle when the machinery beside them is what actually needs fixing.
If your prison room does not match this layout
Deep Dish Dungeon has several locked doors and damaged mechanisms in the prison. If you see only one broken lever near the first cells, use an Iron Rod there and keep moving. If you are still outside the prison, you need the Prison Key rather than rods or Chains.
For the large counterweight door, the checklist is short: three repaired lever mechanisms, three Iron Rods available for the broken parts, Chains fitted to the isolated mechanism, and all three levers pulled. Get those four things right and the door stops being mysterious.