I haven't ever faced this type of algea; but I can relate to a period of neglect and having to overcome the shame and challenge of getting the tank back to good shape.
My battle was caulerpa, which completely overran my tank while I neglected it.
There is no easy fix out of this, and I think BRS's message of 'it took months to get this way, it will take months to get back'.
For me, it was the following:
1.) Consistent water changes (These did not happen during the neglect period)
2.) Using a large hose without attachment -- Then using my thumb+hose to physically suck out as much as possible during each water change.
........repeating this for many water changes.
3.) Adding utility fish (surgeon fish/tang).
4.) Adding a refugium to my sump
It took several weeks(months), but these past weeks have been very rewarding. It is not 100% gone, but it is disappearing day by day now. I no longer need to remove large sections during water changes.
This is chronicaled in my build thread -- including photos of it at it's worst and my acknowledgement that I was going to try and fix this.
Here is the last few weeks.
March 30th:
April 21st:
I have not dosed any chemicals. Just returning(and keeping) a consistent husbandry routine and what I described above.
Whichever method you chose to listen to good luck. It can be beat, but like everything in this hobby there is no quick fix.