Where and how did the Inhumane Society keep Lizzie captive?
The organization had four willing members and three unwilling members. The willing ones were Hangfire, Stew Mitchum, Nurse Dander, and Dr. Flammarion (it was never explained why the latter two allied with him). The unwilling ones were Sharon Haines, Sally Murphy, and Ellington Feint (and technically Kellar too, for a while).
But after WDYSHL, there were only two willing members - Hangfire and Stew. Nurse Dander, and Dr. Flammarion got arrested and sent to the city to stand trial (or so we’re told. I don’t suppose it was confirmed if they actually got there?). That leaves only Hangfire and Stew, the latter of whom is a boy who can’t be much older than thirteen, to imprison Lizzie.
Sharon and Sally obviously weren’t helping to keep Lizzie captive, because the only reason they were helping Hangfire in the first place was because he was blackmailing them with Lizzie’s safety.
And Hangfire and Stew were busy with Hangfire’s plans throughout the series. They both make multiple appearances in SYBIS; clearly, they were not actively guarding Lizzie or whatever.
In WITNDFAON, Sally somehow rescues Lizzie, disguises her, and sneaks her onto the train. She believes that she’ll never be free of Hangfire, that he’ll likely kill her like he tried to, but she’s determined to at least get Lizzie out safely. (Lizzie’s acting skills were a lot better than hers, but I think that can be explained away to extreme nervousness and stress.)
But if Sharon knew where Lizzie was, why didn’t she try to rescue her before, instead of being blackmailed by Hangfire?
She may only have found out recently, being somewhat affiliated with the Inhumane Society. Okay then. So she managed to rescue her - Hangfire was unaware that his hostage was gone.
But who was supposed to be holding her hostage? Every member of the Inhumane Society was on the train. Lizzie was not supposed to be on it.
Lizzie wasn’t in Wade Academy, obviously, since Sharon and Kellar were there. She couldn’t have been in the Colophon Clinic, at least not after it was all but destroyed.
Stain’d-By-The-Sea has any number of abandoned buildings, Lizzie could have been in one of those. But who was watching her to make sure she didn’t escape? After all, Nurse Dander was with Ellington disguised as Cleo to make sure she didn’t escape.
Did Hangfire just keep her drugged with laudanum and locked in a room somewhere without bothering to guard her? Under the influence of laudanum, she wouldn’t be able to escape by herself; and, indeed, she didn’t, until Sally found her.