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