The piece that you twist the paddle of whisk onto has broken off leaving part of it in the paddle. Model number is K5SS and the mixer, while 35 years old is still working fine.

SOURCE: My Kitchenaid K5SS 300 watt standing mixer launches into the highest speed regardless of the speed ....

I would recommend that you first check the Hall effect sensor which determines the speed.  I had this exact problem (but mine occurred with a 600 professional model) and I actually went out and bought another Hall sensor for about $13.  I replaced the original sensor and made sure that the connection between the sensor and control board was nice and snug. My blender worked as good as new! But I was curious to know if my original sensor was defective so I put it back in mixer and again made sure all of the connections were tight.  Guess what?  My original sensor also made my mixer work!  There was nothing wrong with my original Hall effect sensor except the connection between the control board and sensor was not adequate--unfortunately I didn't check the connections before I decided to buy another one. My guess is that the connection between the control board and the sensor probably loosens or comes undone over time from use and vibrations. 
1.  Download the mixer parts and diagrams from the Kitchenaid website to see what the Hall effect sensor looks like. 2.  You'll need a phillips head screwdriver to open up the mixer. 3.  Make sure that the connection between the control board and sensor is absolutely tight and snug. 4.  Also, make sure that the orientation of the sensor connection between the control board and sensor is correct. it's hard to know which way the sensor should be connected because the control board has three prongs and the sensor can attach facing up or facing down. If, after you've made the connections, the mixer still doesn't work.  Disconnect the sensor and reconnect it with the prongs facing the opposite direction and try it.  I actually found the right orientation through trial and error.  By going through this trial and error, I also had my Eureka moment when I realized that the original sensor probably would also work it were reconnected with correct orientation.   5.  This whole process probably takes less than 10-15 minutes to perform.

SOURCE: paddle beater won't push up and turn to release

When you are able to remove the paddle, feel the opposite side of the spindle. It should be smooth. The pin should not stick out the back at all. Carefully, tap it in until it is even with the spindle. That will fix you up, unless the pin itself is also damaged. If so, get a small file and remove any burrs from where the paddle mounts.

SOURCE: My 20 year old classic k5ss mixer overheated and

Hi HAMMERSANDCO, Merry Christmas and Happy New Year! Don’t have to throw your old K5SS away. It may be time for some TLC maintenance on the old mixer that could return it to good working order to run alongside your new mixer.
From the symptoms you described, it sounds as if your control plate assembly and/or phase control board has gone bad (possibly as a result of the overheating…electrical components can be very sensitive to extreme heat). 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 and feel adventurous. 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 K5SS mixer falls within this manual), and select the first file KSM50P.pdf (438.31 KB) listed under Parts List (4) and then check to verify this matches your mixer model.
Turn to pages 6 & 7, titled “Motor And Control Parts”, which provides an exploded view for the disassembly and reassembly of your mixer’s electrical components, along with all the parts. You’ll want to focus on Illus. No. 36, Control Plate Assembly and Illus. No. 39, Phase Control Board
Unfortunately the controller assembly is fabricated on a little PC board which was not meant to be 'home fixed', and my 'expertise' ends there. A local small appliance repair shop should be able to determine if the controller assembly is functioning properly or not, or you can just order replacement parts without the testing.
While you’ve got your mixer this far apart you should check for any other unseemly wear and tear on the on the gears and other components. Turn to pages 4 & 5, titled: Case, Gearing And Planetary Unit for the disassembly and reassembly of your mixer, along with all the parts. You’ll have to remove all the grease from the mixer head and gears. Once de-greased, pay particular attention to Illus. No. 58, Worm Gear Bracket & Gear, including Illus. No. 12, Worm Gear (23 teeth). This is the KA fail-safe gear that is supposed to strip to prevent your mixer going ‘up in smoke’ when it is overloaded.
If you’ve never replaced the mixer grease in 20 years, you’ve be rewarded by KA with a gem of a mixer. Can you imagine driving your car and not getting it lubed in 20 years?
You can order a replacement parts and install it yourself. Besides a replacement control assembly and grease you might consider replacing the gasket, and “O”-ring too. 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 may contain contaminants.
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. The mixers displayed in the links are different KA models, but the principles are the same.

• 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

SOURCE: Sunbeam Mixmaster... where's the model number?

Lift up the head, should be right by where you insert the blades

SOURCE: I need the manual for my K5SS mixer my

Hi WWW_SOCRAFTI, you can easily download both the Owner’s Manual and Parts List manual for your K5SS mixer online from the KA website at: 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 K5SS mixer falls within this model range), and then select this file: KSM50PPWH0.pdf (5156.65 KB) from the Owners Manuals (9) list and this file: KSM50P.pdf (438.31 KB) listed under Parts List (4). Check to verify they match your mixer model. You can save them to your PC hard drive, and then you can print a hard copy if you wish, or just refer to it when you need to use it.
The reason I suggest downloading the Parts List manual is that should you decide to try and troubleshoot and fix your KA mixer, this manual will be invaluable in the disassembly and reassembly of your mixer along with listing all the part names and numbers. Having the correct part names and numbers is critical when reordering replacement parts when your mixer eventually will stop working.
Good Luck with your mixer, I hope this helps. Howard, Burke, VA
If this solution has been helpful to you, please rate it, thanks! -hslincoln