In order to wake Kallixenia you need the Waking Stone from Procne, a woman found in Medea's Grove. When you have the stone, simply click on the Altar of Kallixenia, found in the center of the first part of the ruins.
Kallixenia moves fairly fast, has a lot of hp. She has three different attacks, each dealing both Cold and Vitality Damage.
- Spirit Bolt. Launches five small projectiles, each dealing some damage. Her standard attack.
- Spirit Wave. Short ranged wave attack used in melee combat, deals flat Cold and Vitality Damage.
- Soul Strike. Ranged meteor attack. Calls down a couple of energy bolts from the sky, each dealing a larger amount of flat Cold and Life damage and replenishing the Liche Queen's life.
- Cold Damage
- Vitality Damage (Life Leech doesn't seem help eben against the life leech attacks)
- Chance to Avoid Projectiles seems to work against all attacks (not sure about the wave though).
Try to avoid the meteor attack, as it deals a lot of damage and replenishes the Liche Queen's life. And in case melee combat won't work for you - you can kite her, as all of her attacks are fairly easy to avoid at a certain distance.