The thin wrench I assumed I needed is called a Cone Wrench.
From Chat these links were shared with me.
Overhauling a Coaster Brake Bike
Perhaps the answer I am seeking relates to tightening down the "Cone" and then "locking" it down with the locking nut.
At this point the locking nut and the cone are acting as one nut. That seems to be the focus of my query.
Okay.. I have this 12 gauge spoke wheel and it has a "coaster brake" so that is the way it is.
I found out that "coaster brakes" are a concern and that maintenance and rebuilding them are part and parcel.
I trust now that I have wired the arm of this brake with a much better gauge of wire that I will not suffer the same "spin" that I did this morning.
As I mentioned to a Chat-Mate there isn't an obvious way to secure the arm to the frame because of the modifications I have made to the structure of the rear end. Wire is the obvious quick fix and quick fix is what I need because I am riding 24 miles a day 7 days a week until November.
I may consider a different wheel after this seasonal job is finished.
On a brighter note it is fantastic to ride at 27 mph again and at an easy rpm.
Okay then I welcome input since I look to the membership for advice and this is the way it goes for a few more weeks..
I trust Pirate Cycles is selling a wheel that is not unsafe for MaB'ers