Between the old rotating Dremel tools with safety glasses and a supply of cutoff disks and the new multifunction vibrating Dremel tools there is nothing you can't do! Home dentistry, dishwasher repair, carving a Stradivarius violin out of a full gown standing tree, cutting off frozen bolts, making sushi, etcetera.
Incidentally, the part #9 is actually threaded into the bottom of part #27. Flats on the wash arm should, in theory, hold part #9 steady so that you can unscrew part 27. In my case, the two pieces were frozen in place with lime deposits.
Holding the wash arm and twisting the tower just deformed the plastic parts enough so the wash arm twisted past the flats rather than it acting as a wrench to hold the lower piece in place while I unscrewed the tower piece.
(By the way- Part #7 also has flats on it which allow it to spin when part #9 spins, but the wash arm doesn't actually hold part #7 directly. Those flats just cause part 7 to spin when part 9 spins.)
Maybe, in retrospect, pouring vinegar down the sides of the tower and letting it soak overnight might have softened up the deposits enough to allow the tower to unscrew without carving it apart with the Dremel tool.
The fastest, cheapest, and least aggravating solution however was to just cut off parts 27 and 9 by sawing just above the wash arm and then replacing those two pieces of plastic for about $25.00 and change.