I can't remove the beater

I had this happen on my Kitchen Aid mixer. There is probably a build up of dough/cake mix etc inside the hole where the beater fits into. Fill a mixing bowl or a jug deep enough to fit over beater with very hot or boiling water. Immerse the stuck beater in the water up to where it connects in to the mixer and hold for about minute. Once the hot water has melted the build up you should be able to remove the beater. Hopefully this may work on your brand of mixer.

Do you mean the beater is not touching the bottom of the bowl? AS in there is a gap between the beater and the bottom of the bowl so it is not mixing all the ingriedients?
If you have a Tilt-head style mixer, you can adjust the beater/bowl clearance by turning out the adjusting screw. Place your bowl on the bowl screw cap, Place the flat beater on the agitator shaft and lower the head of the mixer so the beater is at the bottom of the bowl. There should be a 1/16" clearance between the bowl and the beater, if the beater is too close to the bowl turn in the adjusting screw, and if the beater is too far away turn out the adjusting screw. The adjusting screw can be located by tilting the head of the mixer back, and in the neck of the mixer you will see a flat head adjusting screw.
However if you have a bowl lift mixer and the bowl is too far from the mixer there are a few different way to fix that problem. If the bowl and beater are too far apart start by adjusting the adjustment screw. On this mixer the adjustment screw is under the head. Remove the bowl, and beater and lay the mixer on its back. You should be able to see under the head a screw sticking out, it does not have a head, but a little slot in the screw, use a flat head screw and adjust the beat 1/4 turn. Put the beater and bowl back on and test mixer to see if it the beater has a 1/16" clearance. If it doesn't adjust the screw and 1/4 turn. If after adjusting the screw the beater and bowl still have too much clearance use a mallet and tap both sides of the yoke arm up evenly until 1/16" beater to bowl clearance is met.
You can use the same dime test in the bowl to make sure you have the correct 1/16" measurement for the beater/bowl distance.
Thank you,
Colleen
MendingShed.com
800-339-9297

Use a small amount of vegetable oil to lubricate the beater shaft and
the beater. Slightly tap the beater up and sideways with a rubber mallet
until you can move the beater easily along the shaft.

Make sure your Beater can slip in and out of the Beater Shaft
easily. Do not force the beater onto the Shaft if the shaft is bent.
If it cannot be removed easily by hand, use a soft rubber mallet
(not a steel hammer) to slightly tap the beater so it can be moved
up and down the shaft or from side to side. Then you can easily
turn the beater left and remove it from the shaft.

From Kitchenaid Manual:
To Attach Flat Beater, Wire Whip, or Dough Hook
1. Turn speed control to OFF.
2. Unplug stand mixer or disconnect power.
3. Tilt motor head back.
4. Slip beater onto beater shaft and press upward
as far as possible.
5. Turn beater to right, hooking beater over pin on shaft.
6. Plug into a grounded 3 prong outlet.

To Remove Flat Beater, Wire Whip or Dough Hook
1. Turn speed control to OFF.
2. Unplug stand mixer or disconnect power.
3. Tilt motor head back.
4. Press beater upward as far as possible and turn left.
5. Pull beater from beater shaft

Make sure your Beater can slip in and out of the Beater Shaft

easily. Do not force the beater onto the Shaft if the shaft is bent.

If it cannot be removed easily by hand, use a soft rubber mallet

(not a steel hammer) to slightly tap the beater so it can be moved

up and down the shaft or from side to side. Then you can easily

turn the beater left and remove it from the shaft.

From Kitchenaid Manual:

To Attach Flat Beater, Wire Whip, or Dough Hook

1. Turn speed control to OFF.

2. Unplug stand mixer or disconnect power.

3. Tilt motor head back.

4. Slip beater onto beater shaft and press upward

as far as possible.

5. Turn beater to right, hooking beater over pin on shaft.

6. Plug into a grounded 3 prong outlet.

To Remove Flat Beater, Wire Whip or Dough Hook

1. Turn speed control to OFF.

2. Unplug stand mixer or disconnect power.

3. Tilt motor head back.

4. Press beater upward as far as possible and turn left.
5. Pull beater from beater shaft.

Make sure your Beater can slip in and out of the Beater Shaft
easily. Do not force the beater onto the Shaft if the shaft is bent.
If it cannot be removed easily by hand, use a soft rubber mallet
(not a steel hammer) to slightly tap the beater so it can be moved
up and down the shaft or from side to side. Then you can easily
turn the beater left and remove it from the shaft.

From Kitchenaid Manual:
To Attach Flat Beater, Wire Whip, or Dough Hook
1. Turn speed control to OFF.
2. Unplug stand mixer or disconnect power.
3. Tilt motor head back.
4. Slip beater onto beater shaft and press upward
as far as possible.
5. Turn beater to right, hooking beater over pin on shaft.
6. Plug into a grounded 3 prong outlet.

To Remove Flat Beater, Wire Whip or Dough Hook
1. Turn speed control to OFF.
2. Unplug stand mixer or disconnect power.
3. Tilt motor head back.
4. Press beater upward as far as possible and turn left.
5. Pull beater from beater shaft

You have probably stripped a gear in the gear housing. Here is the breakdown of the gearbox:

http://shared.whirlpoolcorp.com/assets/pdfs/literature/Repair%20Part%20List%20-%208212010.pdf

The newer "more improved" mixers use plastic gears and they can't take the beating, no pun intended, and fail.

The good news is that the most likely gear to fail is cheap. Here it is on amazon.com:

http://www.amazon.com/KitchenAid-W10112253-Worm-Gear/dp/B0006HLAVA

http://www.misterfixit.com

It sounds like you have two problems. First, the black marks are usually because of a reaction between the beater and certain chemicals used in dishwashers. Try washing the beater by hand and scrubbing it throughly to remove any residue. If this works, don't use the dishwasher again. You could also buy a plastic coated beater.

It also sounds like the bowl height needs adjusting. There is a screw just under the head for this.

The beater is held in place by a pin and spring arrangement. Lift and twist the beater. If that doesn't work, the pin is trying to come out...and is wedging against the beater. Tap the pin back in a bit, and the beater should release.
Regards, --W/D--

Will it turn at all? Try to soak it. Put a paper towel strip where the beater locks onto the shaft and soak ot with window cleaner. Let it soak for a while. If there's any buildup, this should soften it.