Spitalfields is where London keeps its history least curated. The Huguenot silk weavers arrived after the Edict of Fontainebleau in 1685; their terraces on Fournier Street are still standing and still private residences. The neighbourhood became Jewish in the 19th century, Bangladeshi in the 20th, and is now being gently dissolved by finance-industry money, but it moves slowly enough that the layers are still visible.
Fournier Street is the spine: walk its length slowly and note number 19, which has been a chapel, a synagogue, and a mosque without ever being demolished. Dennis Severs' House nearby requires a booking and rewards it. End at the Whitechapel Gallery, which has been showing serious art since 1901 and is free.