UnderMine

There are 3 main steps to freeing Lillyth, and all occur randomly in Delvemore Dungon. I completed each step on a separate run. You might get lucky and be able to do all 3 steps in a single run.

Step 1:
You will randomly run across a bomb-able rock with an obvious "X" on it. You must bomb this rock and enter the secret area that is uncovered. After a long hallway, you will come to Lillyth's cell. There will be a non-aggressive mage and a knight guarding it. Do NOT attack them. I spoke to them both until all dialogue was exhausted. You may also need to speak to Lillyth through the bars. If I remember correctly, the mage will kick you out because you do not yet possess the key to the cell. This is supposed to happen.

Step 2:
You must find a room containing the gargoyle mini-bosses and defeat them. This is a single, specific room containing 4 gargoyle statue enemies that you must fight at once. This room was not a secret or hidden room. For me, this room appeared on Delvemore Dungeon level 4. It requires a standard key and will appear as a skull on your mini-map (just like a boss). The way that I knew I found it is there were 2 boss skulls on the mini-map. This seemed odd to me so I used a key on the 2nd skull room and it was the gargoyle mini-boss. I completed this fight very late in the game because I originally missed this achievement due to it's heavy RNG components. The fight was very easy for my heavily upgraded character. If you are having trouble, use bombs when they are in statue form. After killing the gargoyles, grab the Master Key off the pedestal. This will unlock the achievement:
Step 3:

You must find the randomly placed "X" marked rock again, like in step 1. Bomb the rock, enter the secret area. Go down the long hallway again and this time, use the Master Key to unlock Lillyth's cell. Speak to Lillyth until she gets teleported away by the mage. This will unlock "A Dangerous Ally" and Lillyth can now be found in the main hub area.

This is my first guide, please let me know if I can improve.

Update with some new info: According to Sherlocked4evr in the comments, you will have to find Lillyth a first time and talk to the guards before the gargoyles will start to possibly show up in your runs. I can confirm I had found her once before and talked to the guards without the key before later freeing her. I just didn't know that was a requirement.

Lillyth can be found in the Delvemore Dungeon. You should hold on to whatever bombs you get on your run when going for this achievement.

In order to rescue her, you'll first have to kill the gargoyle bosses, which appear randomly in Delvemore. If you find them on your run, defeat them to earn an artifact called the Master's Key. You'll also get an achievement for defeating them. It can be a tough fight depending on how many upgrades you have and which relics you found on the run. This battle is one of the reasons I recommended holding onto your bombs; you can use them to blow up the gargoyles when they turn back to stone for their healing phase, speeding up the battle considerably. However, you'll still need at least one more bomb for the next part, so try not to use up all that you have in the battle.

Once you have the Master's Key, be on the lookout for a single stone with a small "x" on it, which will appear somewhere in Delvemore. Blow up this stone with a bomb and a staircase will appear. Go down it, then head up through a few rooms until you see a door being guarded by two people. Simply walk up and unlock the door, freeing Lillyth and earning this achievement.