The woman, in her 20s, suffered multiple gunshot wounds after two male suspects fired seven shots into the address in Cronin Street between 10.30pm and 11pm on December 24.
She was taken to hospital in a potentially life-threatening condition and remained in a critical care unit over the Christmas period.
She is still on a ward receiving treatment.
The child was unharmed.
Scotland Yard has appealed for witnesses to come forward, saying the two suspects fled on foot towards Commercial Way.
The force said both men were dressed in dark clothing, with one wearing a grey hoodie and believed to have been carrying the gun, while the second was wearing a dark hoodie.
Detective Inspector Jordan Gardner, who is leading the investigation, said: “This was a shocking act of violence, carried out at a family home where a young child was present. It is vital we identify those responsible and ensure justice for the victim.
“We know there will be people who saw or heard something that could help us. Even if the information seems minor, please come forward.
“We are particularly keen to hear from anyone who was in the area of Cronin Street, Commercial Way or Moody Road between 21:45 and 23:45hrs on the evening of Wednesday, December 24.”
