How to pull out the motor apart on a model 3b because I can not pull the rest of the brush that's still inside mixer

OK the answer from Ross Keen above is correct as far as it goes. If his way didn't work then a tear down is the only way to fix the problem. I will make two assumptions, Fist you did try to free the stuck brush by turning the mixer on it's side and tapping with your hand or a light weight rubber / leather mallet. Second you have your new brushes in hand, a new motor head gasket and new grease. Why the gasket and new grease? Because if you are going to tear down the mixer you can rip tear or break the gasket, a new one is like 5 Bucks from appliancezone.com or on eBay you can get a gasket and grease in one purchase because the can of grease is like 27 bucks by itself, the eBay buy will have just enough to re-grease the mixer. Now why re-grease the mixer...well. you may not get back to opening the mixer again for like 20 years...yeah really 20 years. And if the mixer is over 5 years old ... just put new grease in.
Ok tear down for an Artisan Tilt Head mixer, you want a nice big work space, cover it with an old towel a beach towel preferably, you need the space if this is a first time tear down for you. You want to put the pieces out in order of tear down to keep things straight for re-assembly. You will need a Phillips screwdriver, a nut driver 3/8th inch, needle nose pliers, a slotted screwdriver, two large #3 screwdrivers to use as a set of crossed pry bars, a rubber mallet, and two punches one large and one small, a long thin head slotted screwdriver. and the new parts. Now go to this site, it is a complete tear down manual for the tilt head mixer from 1978 to today.

http://www.kitchenaidparts.eu/documents/pdf/service-manual/classic-svc-manual-new-email-version.pdf

Good luck and if you need help let us know so that we can FixYa

You will need to replace the brushes once you get the stuck one out. You want to avoid pulling the motor apart if you can, I carefully drill down through the middle of the brush. Note CAREFULLY. You do not want to drill into the commutator that the brushes run against when the motor is running. Once you have your hole drilled you can try and extract it with a hooked pick type tool or with a vacuum cleaner sucking away at the outside you can break it up and it will come out in many pieces. Once you have it out clean up the brush holder with a small flat file and check that the new brush slides in and out freely. When filing be careful not to mark the commutator just beyond the inner end of the brush holder. Brushes are keyed and will only fit one way, look in the hole with a torch to see the key on one corner.