Well, this is interesting! I've learned that a significant inspiration for not only the style of the Planescape setting but the Lady of Pain herself seems to have been a Swinburne poem, "Dolores," or "Notre Dame des Sept Douleurs."
The text can be found here: https://www.poetryfoundation.org/poems/45283/dolores-notre-dame-des-sept-douleurs . I agree with other critics that it's quite long and repetitive -- you can skim it for the style. The lady being discussed here is unlikely to be an accurate representation of the Lady of Pain, unless she's considerably more lustful than most representations have her!
I have passed from the outermost portal
To the shrine where a sin is a prayer;
What care though the service be mortal?
O our Lady of Torture, what care?
All thine the last wine that I pour is,
The last in the chalice we drain,
O fierce and luxurious Dolores,
Our Lady of Pain.
When, with flame all around him aspirant,
Stood flushed, as a harp-player stands,
The implacable beautiful tyrant,
Rose-crowned, having death in his hands;
And a sound as the sound of loud water
Smote far through the flight of the fires,
And mixed with the lightning of slaughter
A thunder of lyres.
Dost thou dream of what was and no more is,
The old kingdoms of earth and the kings?
Dost thou hunger for these things, Dolores,
For these, in a world of new things?
But thy bosom no fasts could emaciate,
No hunger compel to complain
Those lips that no bloodshed could satiate,
Our Lady of Pain.
I was alerted to this by a post from David Prokopetz on tumblr: https://prokopetz.tumblr.com/post/177941348672/theyoungerwhatelydaughter-prokopetz .
He mentions mimir.net there as a defunct site; I don't have a tumblr account, so perhaps someone that does could drop a note replying there mentioning that planescape.com is still trying to be up and active?