It only mixes on slow speed

it is most likely the armature. You can test the coils with a meter set to ohms. http://www.appliancepartspros.com/motor-control-parts-for-whirlpool-k5ss.html

the mixing speed was attained by the use of a potentiometer ( variable resistor ) that would have controlled the power into the board or from the board to the motor
it would be under the speed setting knob
maybe the resistor in use is not the right one and is not reducing the voltage sufficiently to do the job

I am assuming you mean the little black knob on the speed control has broken off. I will post the solution for that problem.
The speed control lever on KitchenAid mixers has a little black knob on the end (in some cases this may be white or gray). The knob is pressed onto the lever, so if the knob breaks off your only option is to replace the whole lever.
KitchenAid has a kit that will include all you need to replace the speed control lever. It even comes with instructions. Here is a link to the speed control lever kit on our website: http://www.mendingshed.com/kaleverkit.html

The lever kit includes the speed control lever (with knob), speed link, spring, instructions and a gearbox gasket.


The instructions will only walk you through how to hook up the speed control lever, so if you also need a little guidance in taking the mixer apart, and reassembling it please use the KitchenAid service manual. The manual will walk you through all of the repairs you may need on your mixer, including how to take the mixer apart and put it back together. Here is a link to the service manual: http://lib.store.yahoo.net/lib/mendingshed/kitchenaidservicemanual.pdf

Remember, The Mending Shed is a Authorized KitchenAid service center, we can help you with any repair needs for your KitchenAid appliances, plus all of the parts we carry are genuine KitchenAid brand.

Thank you,

Colleen
Customer Service
MendingShed.com
800-339-9297

You can get the high speed back if you want to play with those three screws. there is a timing scheme for your mixer and it is set and adjusted by manipulating those 3 screws. The bottom center screw is for speeds 6 8 and 10. The best choice is to replace this switch, about $10 bucks on ebay.

Sounds like the governor has flown apart, you will need to take that black plate off the back, all the wires off it (take a photo first if you think you will need it) a spring at the bottom and 2 screws at the top. Once you get that off the broken governor will be exposed and you can lever it off with a couple of screwdrivers, don't lose the locating pin through the motor shaft.
Part required is
17830 Governor Assembly
Line up the notch in it with the pin you have in the motor shaft and bang it on with a soft hammer so that the notch engages the pin.
When fitting the back plate back on first engage the speed control lever into where it belongs to operate the switch then put each screw in 8 full turns to start off. I put each one in a couple of turns then the other one until each is 8 turns in. Put the spring back on, put the wires where they belong and fire it up on low speed. Move it up to high then back to speed 1. You want the planetary to turn 13 to 15 revolutions in 15 seconds. I let the planetary knob brush my hand as it rotates while watching the second hand on the clock. Turn the screws in to slow it down and out to speed it up, you should move each screw equally. Once slow speed is set lock up the nuts that stop those 2 screws from turning. Job done!

By not working to mean that the planetary gear wont move? or does it just make a weird noise when it is on a low setting? If the gear is completely stripped the unit shouldn't run on any speed regardless if there is mixture in the bowl or not. Since I can't see the mixer I would say that there might be a slight tear in the gear which would only allow the mixer to get stuck on a slow speed where as on a high speed it is running fast enough to skip that particular tooth. I understand that the mixer is brand new but when it comes to mixers you can strip a gear right out of the box depending on the consistency of the batter you are mixing. If the machine is under a year old it is under warranty and you are allowed free service by an authorized repair center in your area.

Sounds like a problem we had years ago. We put too much pressure on the motor by mixing a very thick dough for cookies. It stripped the "gears" and couldn't be used on slow.... it would instantly go to the fastest speed. We had to take it to a reputable appliance store that does repairs on KitchenAid products. I'm still using mine and that has been 15 years ago. It still works great!!

Here is how your speed control works:

Speed Control
The speed control of the mixer is attained through the use of a governor assembly mounted at the rear of the control plate assembly.
The electrical circuit is made and broken by the action of the fly ball governor revolving against the control plate.
When the switch lever is moved to an ON position, the position of the control plate with respect to the governor is changed by the action of the switch lever.
Thus, when the control plate is set close to the governor, a relatively low speed of the motor causes the governor to make or break the mixer’s electrical circuit through the control plate.
When the control plate is set farther away, a greater motor speed is required before the governor starts breaking the circuit.
The action of the governor is such that the speed of the motor will remain constant for a given setting of the control plate within certain loads.
After certain loads have been exceeded, the speed of the motor will drop to meet the torque requirements of the given load.
Speed is controlled by the governor and the control plate in conjunction with the phase control.

TRIAC RMS VOLTAGE
BOTH CONTACTS OPEN - 40 VOLTS
ONE CONTACT OPEN - 80 VOLTS
BOTH CONTACTS CLOSED - FULL VOLTAGE
NOTE: The triac regulates the power the motor sees depending on control board contacts.
A device called a triac is a part of the phase control circuit.
This device determines the amount of power the motor sees dependent upon the condition of the control board contacts.
If both contacts are open, about 40 volts RMS is applied to the motor and about 80 volts is applied when either contact is closed and the other is open.
When both contacts are closed, the triac is full on; consequently, full power is applied to the motor. In this way, the speed is controlled.
If the mixer motor begins to run too fast for a
particular speed setting, one or both contacts open, which cuts back on the voltage the motor sees, thus slowing it up.
If the motor operation becomes too slow, one or both contacts will close, applying the needed voltage to the motor to sufficiently increase motor speed.
This is always accomplished through the triac.
The control plate contacts control the triac, and in turn, controls the amount of power supplied to the motor.

The control Plate for your mixer is PN
W10119326
http://www.marbeck.com/mixers_kitchenaid_parts.html

If your not good with tools / meters or tedious repairs do NOT attempt this yourself.
The service manual for your mixer can be found here
http://www.f-ems.com/download/DownLoad.aspx?DocsID=082009012917

and is:
LIT4177310-C

ship it back to anvil , buy a hobart ,best mixers in the world

OK, you mentioned everything except the brand. I found on some models of Sunbeam mixers that the speed governor would fail due to the control knob screw being loose.

On that particular model, the screw is behind a stick-on label in the center of the knob. Peel off the label and make sure the screw is tight.

If your mixer isn't a Sunbeam, please advise.