I would suspect that the capacitor (or capacitors, depending on model) for the motor is/are the fault.
These can start to increase in ESR as they age and degrade, and thus, since the Equivalent Series Resistance increases, they dissipate more heat.
Power varies with current squared x resistance, if you remember from your school Physics classes.
More heat means burning smell, and smoke eventually, even though the brushed motor appears to be functioning normally. If you look at the capacitors, they are often cracked, some time almost split in two as a result of the issue.
While you have the machine apart, I would also replace the parts listed in one of the many ebay sellers who specialise in this kind of thing. Look for "Kenwood Chef" "repair kit" and make sure you pick the correct kit for your mixer.
You might save a few pennies by identifying the exact component on your machine, but the other capacitors, resistors (and perhaps triac, depending on model), will probably fail eventually, so you might save yourself another tear down, by simply replacing the lot.
This last piece of advice applies to the 700 series, 900 series, KM300 series... in fact, most of the Chef and Master models. Also check the brushes are correctly seated.
If they are worn, or not seated correctly, they will spark, or make intermittent contact, which can cause the motor to pulse, and the sparking can kill the triac. The symptoms of this problem are a "gritty" sound from the motor as it runs, or pulsing high speed, low speed, high speed at random. If the triac fails open circuit, the mixer will be dead, if it fails short circuit, the mixer will run at high speed, regardless of the speed controller setting. If the triac is starting to fail, or overheating due to arcing, the mixer will typically start off acting normally, but get progressively worse, either faster, slower, or pulsing more and more randomly.