After completing Entertainment Magic, talk to the desperate woman at a gambling table in the Entertainment den gives.
Learning the game
You first need to learn how to play Slay the Demon. For that, you just need to play a game against Annette. You don't need to win.
Game tips
If you are willing to increase your Corruption, two tricks make victory easier at Slay the Demon. After the first game, you are allowed to remove Aura's collar to raise her avatar's base HP, MP and class points to very high values (Aura temporarily reaches level 99). If you know Tailwind II, the other trick is to use The Art of Tailwind. This skill is taught by the jailed mage in the Verdeaux's Knights building (second floor). Learning The Art of Tailwind raises Corruption by 1 and allows you to manipulate all dice rolls. Note that changing a dice roll costs MP and the cost is paid by Aura, not her avatar.
One of the NPCs in the den is willing to give you 500 G and an artifact granting a +10 class points at the start of the game. In exchange, he asks for a blessed anti-magic coating. The exchange can help you, but isn't absolutely necessary.
In general, items tend to be important in Slay the Demon. Acid bomb and Demonbane are great for the final opponent. Coatings costs one class point and tend to give higher value than stats for this cost. Recovery items are good, provided they have enough effect, but a random boon can also grant you 50% HP recovery. This boon only appears if your avatar is under 50% HP.
Besides that, multiple builds to your Avatar can make you win at Slay the Demon without too much difficulty.
Your build will also determine what curses you should take when you must receive one. If you can win the next fight suickly, take a battle debuff. If a stat is useless to you, take debuffs to this stat.
Shirleen's deal
When you challenge Shirleen, she'll offer you a deal. Accepting the deal requires 15 Vice, gives 2000 G and fails the quest. Note that you can still challenge Shirleen again to get one of her skills.
If you reject Shirleen's deal, or play a rematch against her, you have to accept a penalty contract if you lose. Shirleen cheats during your first dice rolls, so you get a 1 at each initial stat bonus. You can counter this with The Art of Tailwind. You complete the quest when you beat Shirleen while having rejected her deal. In any case, you can only win one game against her (she refuses to play against you after that). You can thus learn only one of her skills.