Paddle wont turn - KitchenAid 32685 Mixer

Try and get some CRC or similar into the connection so that it can get between the planetary driving shaft and the paddle. Elbow grease and muscle is what is going to free it up. Twist & turn until it frees up. Once you get it off polish up the shaft and scrape the hole clean with a knife or similar. There will be a black build up in the paddle hole. Flour gets in there and works itself into the alloy creating that black crust that locks the paddle onto the shaft.

You have solved this one already.

Is the humming noise the sound of the motor straining to turn without moving (it won't last very long like that) or the sound of the motor turning but not operating the paddle?

Get some CRC into the top between shaft and paddle socket and in and around that driving lug on the side. Then grasp the paddle with a hand on either side and rock (turn) it back and forth as forcefully as you can until it starts to come free. Eventually it will start to move and you will be able to remove it. Polish the shaft and scrape out the hole, it is flour that has wafted in there that has tightened it up, it packs & embeds itself into the aluminium paddle.

Find the screw in the picture as shown and screw it counter-clock-wise which will drop the bowl level down when it is in the raised position and the mixer paddle will not rub on the bowl.
http://www.fixya.com/uploads/images/07F1BE1.jpg

if the paddle is slipping while the agitator shaft is turning then you may have a problem with the agitator pin shaft it is either gone or not engaged

Tip the mixer up and look for a screw on the hinge underneath the top part. Turn this screw counterclockwise about 2 turns and the paddle will clear the bototm of the bowl.

There is a screw that turns about 180 degrees on the stand behind the bowl that fine-tunes the height of the paddle.

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.

Using one of the bowls, filled it with hot water up to the shaft and where the paddle is hooked up. Using a cotton cloth, wrap it around the paddle and shaft, let it soak for 30 to 40 min. After 30 min, twist the paddle back and forth to loosen up whatever got stuck inside the shaft and paddle. Put a little of mineral oil so that the paddle will be easily remove and it will just slide out from the shaft.