I had a bit of a hunch and it seems I was correct.
As Jane seemed to be the one looking after Agnes, I wondered if she was the daughter Agnes thought people believed she had killed. I searched the death records and found she died in 1886. 2 years before Agnes was admitted to the asylum.
Her brother John informed the death as well as petitioned for her admittance. I presume Agnes stayed with him until it got too much for him and his family. Her address prior to admittance was 12 Charleston (a nearby village). If I can find evidence of John living at this address, it seems to be the most likely scenario.