The date of birth fits the 1939 register - the Eliza Ann birth was registered July-Sept 1865, which does fit
I don't think Annie Janes who marries Francis Banks gives a father when she marries, judging by an index on Ancestry. which might count out the daughter of Jessie. He marries Fanny Lawson in 1865, and there's this birth registration - better fits the 4 year old in Jessie's household on 1871
JANES, ANN
mmn LAWSON
Q2 1867 LEIGHTON BUZZARD Volume 03B Page 415
This person on 1871 in Edlesborough is interesting. I can't find a birth registration to match
Thomas Janes 50 ag lab b Eaton Bray
Sarah Janes 50
John Janes 19
Abel Janes 15
Mary Janes 13
William Janes 11
Emma Janes 10
Ann Bones 5 Visitor b Eaton Bray
The same child appears to be enumerated as Ann Janes granddaughter, b Eaton Bray, in 1881