The child "William" who was 2 in the 1871 census at reply #18 may correspond with this death:
Death, Sep qtr 1871
Frederick William H COOPER aged 3
St Olave, Southwark 1d 144
...and with exactly the same reference (same quarter, same page no - they must have been registered very close together if not simultaneously) this may well be the baby "Jane" from the 1871 census:
Death, Sep qtr 1871
Dorothy Jane H COOPER aged 1
St Olave, Southwark 1d 144