Merry Christmas WESTHOFFL! Unfortunately from the symptoms you describe, your mixer has experienced a total transmission failure. You don’t mention how old your KA mixer is or how hard it’s been used, but for your particular model, KA has a plastic gear designed to fail instead of the whole mixer ‘going up in smoke’. And what it sounds like is that this fail-safe gear finally ‘failed’ (stripped) so that the power from your electric motor is not getting to your planetary and the agitator shaft. What can you do?
The good news is that this can be fixed quite easily (OK, maybe a little effort) by yourself, if you are a DITY type person, feel adventurous, and your mixer is not under warranty. You can troubleshoot it yourself. Be sure you have the Parts List manual on hand. If you're lacking one, you can download a PDF version from the KA website: http://www.kitchenaid.com/flash.cmd?/#/page/home.
Click on Customer Care>Locate Manuals & Guides> Enter the 6 character model number, in your case use KSM50P (Your KSM90P mixer falls within this maual), and select the last file KSM90PSWW0.pdf (354.08 KB) listed under Parts List (16) and then check to verify this matches your mixer model.
Turn to pages 4 & 5, titled: Case, Gearing And Planetary Unit for the disassembly and reassembly of your mixer, along with all the parts. Locate Illus. No. 36, Worm Gear Bracket & Gear, including Illus. No. 7, Worm Gear. I suspect this is the weak link in your mixers transmission and causing the problems. After clearing all the grease away, you’ll probably find that is has been stripped.
You can order a replacement gear and install it yourself. While you’ve got your mixer this far apart you can check for any other unseemly wear and tear on the on the gears and other components. Besides a replacement gear, and grease you might consider replacing the gasket, and “O”-ring too (If you’ve noticed oil leaking). You should use a food-grade grease that you can order online (although I’ve seen automotive bearing grease used too – your call). Do not try to reuse any original grease as this should be filled with debris from the stripped gear.
Here are five potential vendors in no particular order: RepairClinic.com; mendingshed.com; searspartsdirect.com; partstore.com; and Grainger.com. You should search for ‘food grade grease’. Prices can vary widely between vendors, so do your comparison shopping.
It can be messy but quite gratifying to diagnose and repair your mixer yourself.
Finally, here are some links to several excellent websites that detail the disassembling/reassembly of a KA Mixer.

• http://www.artisanbreadbaking.com/discussions/kitchen-aid.htm
• http://www.neilcrockett.com/mixer.htm
• http://www.foodonthefood.com/food_on_the_food/2009/04/how-to-repair-a-kitchenaid-mixer-yourself.html