Hmm...
That's a weird one. While petek's right that it has an overall resemblance to a Ham Beach, HB's don't have the power take-off at the front of the mixer head, the way your Kenmore does. That take-off is in the same place it would be for a KitchenAid-built machine. What I'm thinking is that you either have a KitchenAid-built machine with a special, Kenmore-only bowl (Whirlpool, owner of KitchenAid, has had a relationship with Sears for decades), or you have a Chinese-built KitchenAid clone. Some Chinese products copy other brands very closely.
You may just have to bite the bullet and pay the Sears Parts Central price for another bowl. A word of advice - if this is a Chinese-built machine, I personally would purchase a second electronic control unit (the piece where the pushbuttons are) and put it away in case of future failure. Sears is the dad-ratted worst about discontinuing parts and stranding you with an unrepairable appliance. I used to have almost nothing but Sears. Several major letdowns later, I've phased all Sears out of my life. If this is KitchenAid built, then the electronic parts will probably cross over to some KA or Whirlpool part number, and you'll be okay in future.
It is also possible that Ham Beach makes the machine, and they put a power take-off on the Sears version, but in that case, I can't imagine that KA beaters, etc. would fit.