You might speed up the process by ALSO painting Lugol's directly on the breast twice a day with a Q-tip. Rubbing a small dab of progesterone cream on top of it will speed the uptake of Iodine into the cyst. That's the "Funahashi method."
From personal experience, I might suggest you never, ever let a doctor drain a breast cyst. Shrink the cyst nutritionally. Putting a needle into the breast inflames an already inflammatory condition.
You will be fine. I don't know of any cases where FBD wasn't corrected with iodine. You just need to get the right dose and use the companion nutrients. The breasts may get a little tender before they get better.
BluePelican, please report back and tell us how you are doing!