At least 20 fire engines and 125 firefighters sent to burning block, where smoke could be seen for miles.
Begum was one of three east London schoolgirls who travelled to Syria to join so-called Islamic State (IS) in February 2015.
The Met Police appeal comes after a man was charged with the murder of Tiparat Argatu in east London.
Dozens of people attempted to flee following the arrival of the police by scaling fences.
The Museum of the Home felt "extremely compromised" as it did not remove Sir Robert Geffrye's statue despite public urging otherwise.
The child is in a critical but stable condition, police said.
Specialist firefighters are working to safely recover her body.
Four others were injured in the incident on Wednesday afternoon.
Footage of the incident in east London was widely shared on social media.
David Gomoh, 24, died after being stabbed in Newham, east London, on April 26.