A fire service spokesman said the whole of the roof of the museum and library was on fire.
Station Manager Peter Critchell, at the scene, said:
"Flames are coming through the roof and there is thick grey smoke billowing up. Firefighters are still working hard at the scene to try and bring the blaze under control. We now have water jets attacking the fire from the four corners of the building. We evacuated 30 people from inside the museum and firefighters will be making efforts to salvage as many of the artefacts as possible."
More top news
The victim ran along the platform with his friend when the attacker pulled out a knife and chased them both.
Spray cans were found near the bodies on tracks near a south London station.
Some 81% of passengers are satisfied with their journey, down three percentage points from spring 2017.