Women did work. Florence Nightingale being a noteable example.
It's just that the jobs they could do were very restricted.
May I suggest making your character into a sort of Victorian ere P.I.? A female Sherlock Holmes, if you wish? Conan-Doyle actually laid the foundations of modern investigative police work with his novels (a bit like Roddenberry laid the foundation for much technology we see today). Also, the Suffragette movement was just getting started. It would be almost impossible for a female investigator to be taken seriously by the police, which gives you a pretext for doing the investigation.
If you want a "full time" job for her, you could go with a teacher at a girl's school...
My other idea is to have your character know mine, but not know at first that she is the criminal. Maybe an ex pupil?
Hi woegman. Maybe you can be the bumbling PC Plod, or Inspector Corner of The Yard? Hee hee!
Incidentally, the last RP site I was a member of, I had a couple of Victorian RPs going, but totally different stories.