How important is doublethink to the Party's control of Oceania? How important is it to Winston's brainwashing?
I would say that doublethink is necessary for the Party to be able to control Oceania. Without the ability to do doublethink, the people who run the government would surely go insane and be unable to exert control.
The reason for this is that the people who run the government have to truly believe in what they are doing, even though it is a lie. If they could not engage in doublethink, they would be unable to really do their jobs because they would not be able to believe in what they were doing.
This is a very difficult concept to grasp -- just look at how Orwell himself puts it
The power of holding two contradictory beliefs in one's mind simultaneously, and accepting both of them....To tell deliberate lies while genuinely believing in them, to forget any fact that has become inconvenient, and then, when it becomes necessary again, to draw it back from oblivion for just so long as it is needed, to deny the existence of objective reality and all the while to take account of the reality which one denies — all this is indispensably necessary. Even in using the word doublethink it is necessary to exercise doublethink. For by using the word one admits that one is tampering with reality; by a fresh act of doublethink one erases this knowledge
As far as Winston's brainwashing, O'Brien is trying to get him to be able to do doublethink -- to accept that 2 and 2 make 5 even though he knows they do not.