It's the pipe line that's the problem.
Not strictly a 'pipeline' and the BBC drawing is over-simplified. I spent 33 years in various parts of the oil industry and have an understanding of the basics of drilling operations and offer the following.
In simple terms, the mobile drilling unit (MDU) at the surface drills the well via the blowout preventer (BOP), often known as a 'Christmas Tree', located on the seabed. The gap between the seabed and surface components is spanned by the tubular 'riser'. The drill, drill pipe and well casing pass down through through the riser.
The BOP is operated by umbilical lines from the surface but nowadays, in deeper water, these are being replaced by acoustic signalling systems. It contains valves that will shut off the flow of fluids from the well, even if the drill pipe is still in place. In the latter case, the valve rams cut through the drillpipe wall and sever it completely. What kind of control was in use here I do not know.
The pressure in the well is counterbalanced by drilling mud pumped into it and the hydrostatic head in the riser. This pressure must be continuously monitored and the weight of mud adjusted to compensate for pressure fluctuations. Pressure spikes within the well are common as drilling progresses. On this occasion, it would appear that well conditions overcame the precautions in place and the blowback sent oil through the BOP, up the riser and onto the drilling deck where it was ignited.
Efforts to close the BOP from the surface and also via ROV sent down to the seabed failed. This appears to be the result of some kind of internal problem in the BOP which I, as a non-driller, am not competent to speculate on.
When the MDU sank, it took the riser down with it and perhaps the riser is still connected at both ends; I do not know. However, obviously the riser, which is not flexible enough to take this kind of treatment, at the very least cracked in several places and commenced leaking oil from the well.
Thus, if the BOP still defies attempts to close it, the fabrication of a canopy to enclose the riser cracks and allow the oil to be collected and pumped to the surface is a stopgap measure. The only longterm option is to bring in another MDU, drill into the well from the side and pump in cement to seal it.
Hope this helps.
Barry M