Exploring the ruins
This quest is automatically added to your journal after finishing the main quest Hunting a Witch. More specifically, it is added after meeting Keira Metz in her hideout in the forest (M4,1). You reach that location by walking through the portal in her hut. In this quest you must examine the quarters of an elven mage. You can fast travel to the proximity of the entrance to his hideout immediately (picking the right dialogue option during the meeting with Keira) or reach it later. If you choose the second option, you must once again walk through the portal and go south from the witch's hut. After reaching the destination, walk together with Keira inside.
Watch the cutscene in the first room and walk through the portal created by the witch. Once the two of you split, go forward and eliminate the group of weaker drowners (level 4). After securing the surroundings, walk into the water and dive. You must swim to the neighboring cave. Luckily, no one should bother you while you're doing it. After leaving the water you should reach a fork. The left road will get you to the current mission objective faster. The right one additionally forces you to destroy one of the walls with the Aard sign.
Important - During the exploration of the mage's quarters, you can help yourself with a torch or Cat potion. It will make the darkness much brighter. Alternatively, you can keep close to Keira who will light the neighborhood with her spells.
You should reach a much larger cave with another group of drowners. During the battle you can use the green gas drifting above the ground. Igniting it with Igni sign will increase the amount of damage dealt to nearby monsters. At the same time, you must avoid getting too close to the gas yourself as it can poison Geralt and gradually take away your health points. Reach the stairs and find a passage to the next location. There you will meet again with Keira. Deal with some rats that shouldn't pose serious threat to you. The sorceress will ask you to destroy two rat nests, you can do it. It's unwise to waste bombs on that - instead, use Igni sign near these nests. For "saving" your friend you will receive some experience points.
Move on and stop at the place where the first elven mage projection starts. After watching it, go to another cave. Geralt and Keira must deal with some Wraiths here. Yrden sign (magic traps placed on the ground) is very helpful in dealing with those monsters. Follow the linear path (at some points you must climb) and keep away from new poison gas clouds. Soon you will reach another chamber (more experience points).
Walk towards new projection. After watching it, don't try to blindly find new symbols - touching the incorrect ones will get you into trouble (for example, evil versions of Geralt and Keira might show up and you will have to defeat them in combat). Instead, jump into the large hole in the middle of the room. In the water, dive and swim to another territory. Walk out of water and look around for a horse symbol (use witcher senses if you need). Interact with that symbol. Select a narrow path leading to Keira and together use the new passage. In next room investigate the bird symbol. It will activate a new portal.
Defeating the golem
Another room will force you to fight a mini-boss - golem. It's not a hard opponent, but you must watch out for his strong attacks and charges. In both cases dodge and jump to the sides. Attack the golem when he is turned towards the sorceress, by performing a series of fast hits and one strong one. Additionally, Aard sign can be helpful as it can stun the monster for few seconds. You can also use the elemental oil (if you have obtained it's diagram before).
After eliminating the golem, continue the exploration of the mage's hideout. You must reach a new bird symbol and, after that, unlock another portal. Reach the place where the cutscene will start.
Follow Keira. Remain inside the magical shield that she creates, otherwise you will lose your health points. You must protect the sorceress while she is occupied with closing successive rifts (three in total). Eliminate the Hounds of the Wild Hunt walking from the rifts. To deal large amount of damage to them, use silver sword and Igni sign. Additionally, you can use elemental oil (if you have one). Each time you must kill all hounds; only after that will Keira be able to close the rift. After closing all rifts, the game will give you a time-limited choice. If you want to fight for the sorceress affection, pick the first dialogue option (but it's not a very important decision).
Keep going and save your game. Soon you will meet one of the Wild Hunt soldiers - Nithral. This battle is harder than a duel with golem. Once the combat starts, look for opportunities to weaken the enemy. As in case of golem, use the fact that Keira will distract enemy with her magic attacks. After some time, the boss will hide himself behind a magic barrier and will summon some Hounds of the Wild Hunt for help. It is important to eliminate the hounds efficiently as the boss can heal himself at that time. If you eliminate the hounds quickly, the boss won't be able to regenerate larger amounts of health. Important - don't attack the protective barrier, it won't work and you will get yourself stunned.
Further part of the duel is quite similar. Weaken the Nithral with your attacks and quickly dispose of new hounds summoned after the boss hides behind the magic barrier. Sooner or later you will defeat Nithral. Important - the boss leaves a good sword (Wild Hunt's warrior's sword) and a diagram.
Exploring the last chamber
Reach the last projection and watch it. You will receive The Ladies of the Wood book from Keira and a new main quest Ladies of the Wood will unlock. Remain where you are and activate witcher senses. Examine the herbals and notes. Most important thing to check is the secret exit from the room. Interact with it and listen to another conversation with Keira. She will give you The Eye of Nehaleni. This item is used to see through illusion. Once again check the exit to remove the illusion.
In the next corridor Keira will ask Geralt for help with finding a magic lamp. You can either decline or accept her proposition. The latter will activate the side quest Magic Lamp. No matter your decision, current quest will end and you will receive some experience points. In order to leave the mage hideout as quickly as possible (if you don't want to help the sorceress) pick the left passage, it leads outside (at the end of the walk remove another illusion).