In the 1861 census I found them eventually, indexed as "Burth"
RG9/73 folio 7 page 11
24 Paddington Street (many families at same address)
Ann Burke, 45, widowed, laundress, b. Hertfordshire
Thomas, 14, son, light porter, b. Marylebone
William, 12, son, errand boy, b. Marylebone
Sarah, 10, dau, scholar, b. Marylebone
Elizabeth, 8, scholar, b. Marylebone
There is a possible death of an Ann Burke, aged 52, in 1867 in Marylebone (Freebmd).
In 1871:
RG10/171 44/32
22 Barne Street, Marylebone
Thomas Burke (Burks)?, 25, married, glass warehouseman, b. St Marylebone
Ellen, 34, wife b. Ireland
Anne W, 8, daughter b. St. Marylebone
Eliza, 4, daughter b. St. Marylebone
Sarah, 20, unmarried, sister, b. St. Marylebone
Ellen, 2, daughter, b. St Marylebone
(it is not clear whether Ellen is Sarah's child or another child to Thomas and Ellen).