KitchenAid stand mixers rarely become stuck; if your does, do some simple troubleshooting: Unplug the mixer and plug it back in. Wash the gear knob thoroughly to remove any debris that may be causing it to stick. If you've been using the mixer, allow it to cool for 30 minutes, and then try again.
KitchenAid hand mixers have gears that can become stuck over time so that when you turn the machine on, the knob is difficult to move or the mixer fails to turn on. In some instances, the mixer may work on low speed, but fail to operate when you move it to a higher speed. KitchenAid hand mixers are a non-repairable item, according to KitchenAid customer service, and your best bet is probably to buy a new hand mixer.
KitchenAid mixers are covered by a one-year warranty. If the mixer is still under warranty, contact KitchenAid at 1-800-541-6390. KitchenAid will either replace the mixer, or if it is a stand mixer, pay to have it repaired. If you choose to have it repaired, service must be performed by an authorized KitchenAid service technician.
You will be responsible for the cost of any repairs if your KitchenAid stand mixer is no longer under warranty. Contact KitchenAid's customer service number for information on authorized technicians in your area.