Starting the Cure of Madness
Astrid will ask you to find Cicero's Journal for a clue to were he might have gone to. It is very easy to find as long as you have the quest selected. Afterwards she will give you Shadowmere which is a very powerful horse. You can use it to train weapon skills because of its fast health regeneration and high health. The objective after you have left the Dark Brotherhood Sanctuary is to go talk to Arnbjorn, Astrid's husband. He will tell you that Cicero went into the Dawnstar Sanctuary, but he is wounded. The door to the Sanctuary is right behind Arnbjorn and looks similar to the other Dark Brotherhood Sanctuary. The password is "Innocence, my Brother." which will only be an option once you have started The Cure of Madness.
Inside the Dawnstar Sanctuary
There are a number of traps in the Dawnstar Sanctuary. The first is when you are crossing a hanging bridge. Spikes come out of the wall a number of times, just wait until they stop and you can safely resume your journey. The ghosts are easy to kill using stealth and a bow and arrow. If you have a high sneaking skill you only need to start crouching once you get near the enemies. If you don't be sure to crouch around corners and whenever an enemy is in sight. Use a bow and arrow to kill the ghosts. The best bow is a deadric bow. You can find a guide to making the best armor in the links to the left of this post.
Whenever you see oil on the ground, which is a purple liquid on the ground, look around for something to set it off with. You can burn the oil by using magic, but usually there are other means around. Be sure to utilize these traps to your advantage.
Killing Uderfykte
Uderfykte is a troll inside the ice caves in the Danwstar Sanctuary. He is easy to sneak up on so be sure to kill him using stealth. I found that he didn't do very much damage to me, but I do have armor that has an armor rating of 1000 and around 500 health. If you are having trouble bow and arrow and stealth are your best options.
Killing Cicero
You have the option to let Cicero live. In that case I presume that at some stage it is going to bite you in the ass, but it will get the job done and is a good way to go if you are struggling to kill Cicero because, even though he might not look like it, he will put up a fight. I chose to go the safe route of killing him. Again he should be easy to kill as I had no trouble. After you have killed Cicero you have to return to Astrid to receive the nest quest.
