I found an interesting Joseph Barnett and Caroline in 1891 so I automatically thought of Maxwell.
There is a possible hit with a Henry Maxwell, lighterman, 'married' to a Mary Ann living in Shadwell on Red Lion Street in 1881 where Joe lived and died. He was born in 1844 in Bermondsey but its only seven years off in age to our Henry.
There's a family tree on Ancestry with Mary Ann Steers/Maxwell. She died in 1908 so she's not our Louisa. She's a widow in St. George in the East in 1901.
She and Henry W. Maxwell are not found in 1891 and there's no date of death for Henry so I'm guessing he could still be in the running as Caroline Maxwell's 'husband' if he left Mary Ann. A lighterman and a "night watchman" are not that far apart, are they?
Estimated birth year: abt 1831
Relationship to Head: Head
Spouse: Ellen Maxwell
Where born: Ireland
Civil Parish: Gillingham
Street address: 64 Fox St
Marital Status: Married
Occupation: Night Watchman
Registration district: Medway
Sub-registration district: Gillingham
Ellen Maxwell 49, Accrington, Lancashire
John A. Maxwell 18
Francis Maxwell 12
Samuel Maxwell 10
William H. Maxwell 7
It is actually the little red-haired girl, Kate, who said she heard Mary singing in the November 11 article:
"The girl with the red hair did not think it wonderful that no one had heard any sound of the murder. Someone was always drunk and yelling in Miller court, and she rightly guessed that a woman being beaten would make as much noise as one cut up, so that the murder would not be noticed. For her part she was sure to imagine murder in every direction now. She had strong mind, however, had not had any beer and did not cry. She knew positively that Mary Jane was alive at 1 o'clock, for at that hour she had heard her singing "Sweet Violets" to whoever was in her room. "
Perhaps they didn't take Kate's statement at the time of the murder because she was too young.
The friend of Mary Kelly made Kate "swear an oath" to stay with her that night, so she can't really be her daughter. Since she lives directly across from Mary's, I'm thinking this is Julia Venturnay. Cox lived further up the court.