To call Cinderstorm to fight him you have two choices. One of these choices is to solve the puzzle in the room yourself or you can use the NPC that was comes out of the cage.
If you solve the puzzle yourself, after Cinderstorm is dead a chest will appear in the back of the room. This chest will give you a Potions Cache. The reason why you can do this mainly is because the Potions Cache has a chance of dropping a rare companion. This companion only drops from the Potion Cache's your recieve in Caducius Rise. Also doing the puzzle yourself also gives you an extra plaque of achievement.
To solve this puzzle I worked for a bit one day to figure out the best way to solve this puzzle so here it is.
In the diagram hit the buttons for the puzzle in the order that I have them marked on the diagram below.