I thought the Jane=Kate theory had been based on the fact that Jane had been found in London in 1871 and until this thread Kate had not, so it was thought/hoped they might be one and the same?
On the other hand, a Jane Adams marries John McAuley, Dec qtr 1873 Medway, and turns up in subsequent Kent censuses with him showing a birthplace of Melksham c1847-9, and a clutch of children with mmn Richards.
E.g. Mary Alice McCauley, mmn Richards, Sep qtr 1874 Medway.
So I take it all back

. Looks entirely possible that Kate was really the daugher Jane belonging to that Melksham family, and Florence was their granddaughter rather than their niece.
No doubt the 1873 marriage certificate would be illuminating as to who her father was.