Cody,
The Kock Pouch we are referring to is a type of "continent ileostomy." It is frequently called the k-pouch because it is used for similar conditions as the j-pouch. (And I think because no one wants to risk not knowing whether to pronounce it "cock" or "coke" :-)
The procedure involves making a pouch from the small intestines to store fecal matter after the colon is removed, like the j, but that is were the similarity in design and function ends. While the j is connected to the rectum to simulate normal function, the k is evacuated out a continent value in the abdomen. Unlike an ileostomy which has an "incontinent" stoma requiring an appliance or bag of some sort, the k-pouch is designed using a "continent" valve that closes (only a small absorbent patch needs to be worn over it.) During evacuation, a catheter is inserted into the valve to open it and let the contents out. The other important detail is that most often the entire rectum and anus are also removed and the poucher ends up with what is colloquially called a "barbie butt."
In my case, I may have continued to battle my pouch problems with the j, but rectal problems and leakage complicated things further -- so the removal of that area was my primary motivation. The side benefit I realized, that I hear others also mention, is that k-pouchers can sometimes reduce their incidence of pouchitis. Some think this is due to being able to keep things "loose" -- you aren't trying to thicken your stool because there is no longer fear of leakage. And some cite the ability to irrigate the pouch being beneficial for cleaning out the organisms that might cause inflammation. However, pouchitis diagnosis alone is not usually the immediate reason for this type of procedure, due to the inherent surgical complications, possible continuance and the availability of alternative therapies. For more info, look up SKN69/Sharon's posts -- she is one of the most helpful and prolific posters here and has had a k-pouch for decades.
I hope you this helps and answers your question clearly enough :-) Also hoping you find a good doc to help you with your situation. In addition to the former CC doc above, I've also heard lots of good things about Janet's Dr. Kiran -- who was also formerly at Cleveland Clinic. Though he's a colorectal surgeon, you might be able to get a lead from his office on which GI doc to see there. I understand what you're going through and know it's hard to maintain quality of life like you are... praying for better days ahead!
-- Jen