Here are 2 KitchenAid service manuals
This service manual is not and exact match for the KSM150 but the internal motor control parts and gears are in the the same locations and the troubleshooting procedures are the same: http://www.kitchenaidparts.be/centers/servicemanual/mixer.pdf
And the bigger model service manual with nice to know info:http://downloads.mendingshed.com/KitchenAid_Service_Manual.pdf
These should help you remove the gear. Having said that... here is what may have happened but it is going to be a stretch for a limited use mixer. There may be some wear on the bevel gear shaft and it is catching on the bearing. Before you go hammering on it get yourself some emery cloth / sanding cloth and PUSH the bevel gear back to it's installed position and spin the gear while gripping the emery cloth. Your attempting to remove any ridges on the shaft so that will slide out of the housing.
When it comes time for BRUTE FORCE... make sure you put a small piece of wood at the end of the motor shaft. (not in the books!) I have ruined a motor driving the bevel gear out and it smashed into the motor shaft. (something you do ONCE!) Ever since then I always put a piece of wood between gear and motor when Hammering. The bearing may come out with the bevel gear too. It is almost impossibe to use a bearing press to remove this bevel gear but you can use a C-clamp if your carefully positon it. (
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