Ok, so if you don't want to have to go through the tower to get the key for that quest, then this is what you have to do.
1. Go to Mooshu(yes mooshu). 2. Go on the street where the Lili Ma is. 3. go towards the area where the Fire Exclusive boss is. 4. As you walk down the path to get to where he appears at for fire wizards, you should get the message "Goal complete: collect key". 5. go back to DS, talk to the side quest giver there(forget name, been a while, sry), and you've completed that quest and can do the next one after that if you choose to.
NOTE: I found this out while I still had the quest, I was helping a friend with that fire exclusive boss, when the battle ended, I fought the single enemy there a couple times, then I headed down the street, and the next thing I knew, I completed that DS side quest, and no, I didn't know about it until I witnessed it myself, and the only one that I know of who also knows this is a friend of mine, who was also working on that quest, but not at the same time as me.