Hello Iym'zebey and welcome
I agree with Shir'le and Irennen both. It can be a lot of different things if you/ your DM wants it to be. As for a trial, I think as Shir'le said, an appropriate task may be to defeat a Yochlol or perhaps a monstrous arachnid servant of Lolth like a Drider, or Drider Wizard. Another worthy task would be rescuing an underdark drow that is trapped by the oppression of Lolth. Not converting them, or even telling them of the world above, but protecting them, guarding them as you escort them to a place of safety on the surface. Doing this, especially unasked, would definitely be a deed worthy of a Darksong Knight.
While looking up info I found this old post by Talwyn. viewtopic.php?t=1795
You may already have most of this info, but I thought the list of turnable creatures would also serve as a list of the kinds of creatures that you could defeat to earn the Dark Maiden's favor.
Another thought, as far as the actual ritual, or ceremony of becoming a Darksong Knight, might be to draw from something real world. One lost tradition of becoming a Knight was the symbolic breaking of chains (this represented actual chains that others had work in the past) and then being presented with the sword that did the breaking. There's also the tapping of the sword upon a knight's shoulders, you could play with this and have another knight that is doing the 'dubbing' dance around the one being dubbed in a circle with a sword, then gently tapping her with the sword on her brow as she finished the recitation of a prayer the the Goddess. There's also a tradition of slapping the knight after they've made their oath, which is to remind them of it. This may be a bit too harsh for Eilistraee, but it's worth mentioning.
An even more important and fun topic is exactly "what" you are swearing when you become a Knight of any kind. It's usually the same stuff, but a favorite oath of mine from an even favoriter movie is:
"A knight is sworn to valor,
Her heart knows only virtue,
Her word speaks only truth,
Her blade defends the helpless,
Her might upholds the weak,
Her wroth undoes the wicked."
You could toy with this a bit, though I think Eilistraee would be most happy with it as it is, but maybe change virtue for love and truth for joy?
It also doesn't have to be all ceremony either, it could very easily be far more personal, simple and intimate. You could also go full on old school battlefield promotion with it, wherein you aid a Darksong Knight deliberately and because your sensibilities align with their philosophy in a way that helps them accomplish their goals or saves their life or (even more likely) the life of an innocent. In return, perhaps sensing your characters desire to become a Darksong Knight, they bestow the greatest honor upon you they are able, and knight you then and their. <--- This...may or may not be my favorite.
I hope this helps, I also hope it wasn't overwhelming. I do go on
Have a wonderful day.
Let not the biting, callous and sometimes cruel words of unenlightened and uncaring minds sway your heart from your passions. For it is your passions that are the language with which your soul speaks to the world.