Arthur, I am using an electrolux like the one Jeff has pictured on his site, so I am not really sure how long you will need to mix with your Kitchen Aid. My guess is that you may even need to mix for longer than I do. I will tell you what I do, and you can try adapting it to your mixer. It should work, as the point where I stop mixing is based on a visual clue.
Start by bringing the dough slowly up to the hydration where it is just about to turn into a ball that just collects around the beater. I don't know about your KA, but my electrolux gets to this point somewhere around 63%hydration. If I take my dough any lower than that while it is in the mixer a ball forms an no more mixing is accomplished. So, take it right to the point where you see it want to start balling up. You can dump in 50-75% of your flour fairly quickly pretty much right at the beginning, and just slowly add the rest until you see it start to get to that ball forming hydration. I usually take about 10 minutes in this initial mixing stage to add the majority of the flour.
Once your dough is just about to form a ball around your beater you can walk away and just let your mixer spin the batter for roughly another 10 minutes. There will be a point where the dough starts to change. It is very hard to explain, but you will see it sort of start to harden up, almost as if you had just added more flour, but you didn't. It just sort of starts to go smooth. The change is minimal, but you should notice it if you watch carefully. At this point stop the mixer and pour some of your remaining flour over the dough ball. Your dough should still be very soft and easy to manipulate. If it is not easy to manipulate you did mix it too long so try less next time. I sort of dredge the dough ball in the flour to fully coat it, then as gently as I can I fold the dough over on itself trying to incorporate the final flour from the recipe. Once the dough gets a little sticky again, I coat it in more of the remaining flour and repeat until it is all used up. I think that if I had a professional mixer I could have just added all the flour to the mixer and I would not have to do this final step. I would stop mixing right when the dough started to go smooth, as I think this is the point where the dough is essentially done. Since I am using a home mixer I have to improvise here by adding this last hand kneading step, so I am very careful not to overwork the dough. Just use gentle folds here, no pounding or classic hand kneading techniques here. The last bit of flour is hard to incorporate, so if your dough starts to harden just let it rest for a short time (less than a minute) and it should soften up.
Once all the flour is incorporated let the dough rest for 20 minutes, and fold the dough over on itself a few more times. You are done.
As far as total mix time we are probably talking 25 minutes here. This is more than I ever used to do, and my dough is now the best it has ever been. I no longer need to add American flour, oil, or use an autolyse to get dough that performs to it's maximum potential. Even at 60% hydration the dough stretches out easily never even hinting at a rip. You would think it is 65% hydration by the way that it stretches. Also, I am able to use less bench flour, because the dough actually is quite dry. I get huge voids, and the dough is crispy on the outside, while still staying soft and tender and moist on the inside because of the quick bake.
I haven't tried mixing longer with my American flours yet, but my guess is that I have been undermixing all my doughs.
On to the next level.