The reason that I could not find Kete Ackland in the 1871 census was that she married in 1867, a few months after signing the petition. Her husband was an attendant at the British Museum, Mr Berridge, and she had one daughter, who became an elementary school teacher.
I was intrigued because at the time of the Petition, Emily Davies was also organising the London Association of Schoolmistresses. One of the many members was a Miss Berridge, who taught at Queen's College London. I have not found any connection with Kate's husband, but this could be a possibility... Teachers employed in a large successful school such as this might support the petition, but want to avoid the publicity.