The 35-year-old subcontractor, who does not want to be named, broke his left leg and several bones in his foot in the incident in March 2012.
His fall was two-staged as the unguarded edge was above a plant room.
He dropped into the room and then through a riser duct for an air-conditioning system, where he fell a further four storeys. The total distance was more than 14 metres.
The metal sheeting at the bottom of the duct partially cushioned the impact and may have saved his life.
Westminster Magistrates’ Court heard that the injured contractor was in a loft overlooking the plant room after taking an alternative route from the roof area of the building.
There was no edge protection in the loft to prevent a fall, and the riser duct for the air-conditioning was exposed because a sturdy cover had been removed, leaving just a plastic sheet to keep out dust.
HSE told the court that had the edge been adequately protected, as was later achieved using scaffolding, then the incident could have been avoided.
A properly secured cover for the duct could also have prevented the initial one storey drop from becoming any worse.
The District Judge also heard that Beck Interiors was aware of the need to improve its management of work at height risks, having received written advice from HSE officials and independent safety advisors following earlier visits to this site and others.
The company, of Cox Lane, Chessington, was fined a total of £20,000 and ordered to pay £13,365 in costs after pleading guilty to safety charges.
After the hearing HSE Inspector Stephron Baker-Holmes said: “This was a serious incident that could have ended in tragedy. It is astonishing that the worker plunged so far and escaped more serious harm, and he is incredibly fortunate that the metal duct broke his fall.
“This case highlights the need for principal contractors to proactively manage work at height risks, and to take appropriate action to prevent or mitigate falls.
“The company expressed regret, but asserted that the injured person should not have been in the loft area. However the fact is that others were also accessing the loft, and the risk of falling should have been adequately controlled.”