Each individual level of the dungeon is only one map region, so there are no exits to the north, east, south, or west.
The quest objective has 5 sockets for inserting the Eye of Sertrud amulet. If you own fewer than 5 of these amulets, there is nothing to be gained from the dungeon besides randomly generated loot. Note that amulets of quickening will transform into Eyes of Sertrud after being worn for a random length of time. These sockets are all on the same level of the dungeon, but may be randomly between the 13th and 20th floor.
Unfortunately, there is no way to tell how deep you are at any given time. If you wish to track this information so that you aren't lost after a dimension teleport, you may wish to drop a varying number of coins on the staircase to indicate which floor number you are on.