The only problem with the KitchenAid micer Service Manuals is that they do not address any models prior to those which have the solid-state speed control. The K5-A used a wire-wound voltage divider which is no longer made. The on-off switch was also very different.
There were two types of these switches. The first version was a toggle switch with a forked toggle. A pin on a lever engaged this toggle and turned the motor on and off. The mounting position of the switch affects how well the turn-on function works.
The second version used a "micro switch" which was also lever actuated.
These almost never need adjustment.
Therefore, I assume that your mixer has the first version of the switch.
You will need to open the mixer motor housing up. This is described in the Service Manual. But, since current mixers use an entirely different switch, it is of no help there.
Anyhow, once it is opened you will see the switch and it is a matter of loosening the nuts which hold it to the bracket and adjusting the switch back and forth (or in and out) until the toggle if flipped when the speed control is moved from off to the stir position.
If you would like a photo of the assembly, there is one on the Yahoo WACEM group. Search for "K5-A Wiring."