Great question, and I am looking forward to other people's answers because I too am not totally satisfied with any of the following methods that I have used. And, like you, I don't want my slaw look like the leftover chads from the 2000 Florida election re-count.
To get good-looking slaw requires technique that I have not yet figured out, no matter what tool I use. In particular, which way do you slice the cabbage, and how do you avoid really long pieces (a problem when cutting by hand with a knife), and how do you avoid having the outer leaves come out as big flat chunks?
So, here is what I've used over the years, in no particular order.
1.
Knife, by hand.
2. My 1970s
Rival "Grind-O-Matic," using its slicing cone. Here is a pic showing a similar model to what I own. You can buy modern versions of the same thing:
The
Kitchen-Aid attachment you are referring to is basically the same thing. You have to get the cabbage cut so that it fits, and the outer leaves can definitely be an issue, because the blades "prefer" the stiffer inner core leaves.
3.
Mandolin. The cabbage has to be cut into smallish chunks, and you have to wear a cut-protection glove, but this can work pretty well, and is definitely faster than using a knife. Here is the mandolin I use:
Swissmar Mandolin
This would probably be what I'd use for your chore.
4.
Food processor. This too requires that the cabbage be cut into small chunks. Also, my 1978 original-style Cuisinart has a really narrow feed tube, so the chunks end up really small. This is my least favorite way to go, but a more modern food processor that has a much bigger feed tube might work OK.