I think it's unrealistic to think the microscopic plastic in the system can be fished out again. It's just too small in a too vast area and at the end I'd say, the bigger problem is whether more plastic is put into the ocean. In case it stops, the microplastic will disperse in the gigantic mass of the oceans eventually (a bit like the oil from the spill in the gulf of Mexico; surprisingly, that dispersed well too).
Therefore it's A) keeping the plastic from reaching the ocean. Maybe built a waste incinerating plant on a ship and bring it where the garbage is (usually coast/islands in SO-Asia) and B) getting rid of the present plastic carpet and there - as I describe it in the article - I would simply blow it up. Problem solved ;-)