I'm not the sharpest tool in the shed, but I am a driller. Although, the deepest hole that I ever drilled was 300 feet, I have an idea. I hope that I don't embarrass myself. I suggest dropping a large diameter (4' to 10') casing to the sea floor, over top of the well. I know that it will take a long time to weld the casing together, not to mention finding a mile of it, but hear me out. By the way, my idea can't be all that much worse than using golf balls and rubber to plug it.
The weight of the casing will be enormous, so when the weight of the casing has reached 1/2 the capacity of the crane lowering it, start adding air bags to offset the weight. Once the casing hits bottom, use the submersibles to start releasing the airbags. Hopefully the weight of the casing will make it sink deep into the ocean floor and oil will be diverted to the surface. Upper airbags can be left to help keep the casing upright and relatively stable. The oil could then be loaded into a barge or ship and removed. Tremie pipes, or grouting and flushing pipes could be installed on the entire length of casing to jet the casing deeper in to formation at the ocean floor and then used to grout (cement).
Head pressure in an obstacle, but will basically only be the weight of the oil inside the casing (don't forget to subtract the weight of the outside sea water from that, and the added weight of up-hole velocity restriction (line loss). This approach could possibly make it more difficult to stop or shutoff the well, but it may make is easier since there is a clear path to the wellhead.
I have dropped 16" casing into a well and welded it together. It can take two guys .5 t. 75 hours, so I know that this will be tough, if not impossible. Just something that I was thinking of on the way home from work. I'm sure that the rocket scientists have already thought of it. Not the first time that I thought that I invented something that has already been invented.