LFB said most of the roof and a part of the fifth floor of the building caught fire.
The spokesman said: "We are doing salvage work on floors up to the fourth floor."
It added that 130 people had been evacuated from the building before fire crews arrived.
Parts of two other adjacent buildings have also been evacuated, LFB said.
Three 2.3 stone (15kg) propane cylinders were found on the site.
Fifteen fire engines continued to tackle the blaze on Wednesday night.
A Tribunals Service spokesperson said: "We are at present trying to contact those who were due to have hearings in the building tomorrow to advise them not to attend."
Police have cordoned off Chancery Lane at the junction with Fleet Street and stopped traffic from entering.
Jasper Pandza, who is based at the Maughan Library at King's College, said: "The fire spread very rapidly, one student said it was just a little fire on the roof and 10 minutes later it was this blaze I photographed.
"The library building was luckily not evacuated."
Chancery Lane is an historic area, with some parts dating back centuries. It derives its name from the historic High Court of Chancery.
The area is predominately occupied by the legal profession and consulting firms.
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