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As you say there is no death for an Annie King in the time frame in London. The only one I can find of the right age is this one in Kent
Annie King age 21, September 1/4 1901 registration district Malling Kent vol 2a page 433.
Did Annie have any connection to kent?

Edited
FreeBMD says Malling kent which is about 30 miles South east of Lambeth but there is also  a Malling in London which is about 7 mile East of Lambeth

Edited (again!) should have read your post properly 

The 1841 census shows William with his children the youngest of which is Mary age 3. On the GRO index Mary Ann Gill birth registered Dec 1/4 1837 registration district Stepney vol 2 page 303 .Mother's maiden name Bromley. So it looks as if this is your family
HO107 piece 713 book 2 folio 39 page 5

Max Boyland
Death Age 83
Birth Date 7 Dec 1920
Registration Date Oct 2004
Registration District South east Hampshire
Inferred County Hampshire
Register Number F50
District and Subdistrict 501/1
Entry Number 76

There is no birth registration for a Max Boyland in Dec 1920 or March 1/4 1921 but there is one for a Sidney C M Boyland mmn Crossley in the Portsmouth registration district March 1/4 1921. And another for a Roland M Boyland  mmn Story reg dist Portsmouth Dec 1/4 1920.
There is a tree on ancestry which has the death in 2004 for Max Boyland belonging to the birth of Sidney C M Boyland. On FreeBMD there are two marriages for a Sidney C M Boyland both in the Portsmouth registratio
There must have been a divorce from the first marriage as the wife appears to have remarried in Dec 1/4 1950 registration district Portsmouth.

I am having difficulty finding Mary F Shaw in the 1861 census and also Sarah Walter.
The 1901 census is an interesting one with the relationships between all the members of the household not being correct.
1901 RG13 piece 49 folio 159 page 84
No 2 Bolinbroke Rd??
Julian H Gonnet   44 Head Chef de Cuisine born France, Foreign Subject
Elizabeth A Gonnet 24 Wife Bayswater
Julian Gonnet 16 Son Breadmaker born Islington
Francia Gonnet 13 Daughter, Islington
Angeline J Gonnet 12   Daughter, Islington
Jeanne M M Gonnet 8 months Daughter, Hammersmith
Herbert W Sillifant 39   Son, Single, Merchant's clerk CC Islington
Jane L Sillifant 67 Mother Widow Madeley Shropshire
Sarah A F Walter 78 Single, Aunt, Madeley Shropshire
Annie Rampton 21 Servant Single

Julian H Gonnet's first wife has the maiden name Kneip according to the son Julians birth registration on the GRO index. His second wife Elizabeth's maiden name is Bush according to the birth registration of the 8 month old daughter Ivonne Madeleine Marie.
In the censuses the relationships of the members of the household are supposed to be to the head of the household but Herbert W Sillifant age 39 can't be the son of Julian Gonnet age 44.
Jane Sillifant is according to the 1881  census, Herbert's mother and she was a Jane Louise Walter who married an Edwin Sillifant in 1855 which makes her a relative (sister?)of Sarah's and Aunt to Herbert. The taker of the census got it all mixed up, so is there any actual relationship between the Gonnets and Walter/Sillifant families. Maybe they are just living in the same house.