Well that's a lot better. 12 X 12 is for a fireplace.
Knowing the fundamentals now you can see how much more heat is necessary to leave up a flue to keep flue gas temp above 250* to the top. A taller chimney will create more draft as well, but requires more heat to do so since it cools at it rises. His 12 X 12 cools and needs more frequent cleaning or he is leaving so much heat up, the stove can't heat as large of an area as it should. Draft is the strongest at flue collar on stove and strength goes by the average flue gas temperature and differential temperature outside the flue. So when allowed to expand and cool it is very detrimental to overall draft. The smaller the stove, the more critical heat loss becomes. He would go through a lot less wood with a liner the correct size. Many times people have smaller stoves connected to large chimneys that can't heat the indoor area they are trying to heat, and when they understand the basics, reducing the size of heated area in the chimney they don't need, the stove heats up properly and drives them out.
I would look for a larger stove. Your height and square footage is too much for your Grandma. BTU output goes by the surface temperature of each square inch of surface area. So running a hotter stove makes up for the lack of surface area. But that is shorter duration fires. For coals in the morning you'll want a larger firebox. Grandpa not only has more surface area, it holds more wood for a longer burn.
Make sure you have no air leaks into your chimney flue now. Any leak like a poor fitting clean out door allows cool air to leak into the low pressure area of flue cooling the flue and slowing draft. Spark screens at the top clog since cold outdoor air can condense water vapor allowing particles to stick on screen. Make sure any connector pipe elbows are clean, it's acting like a restriction somewhere. Is the BK a CAT stove?
Also make sure the pipe where it goes into chimney, called the breech, isn't pushed too far into the chimney blocking off the flue gasses. The thimble and pipe should be flush with the inside of flue wall. I assume the flue liner is clean.
I see you're up there in elevation 6 to 7000 feet too. The higher you go, the less atmospheric air pressure you have pushing into the intakes, so like a normally aspirated engine, power drops with altitude the same as less air getting into stove.
Was your local fabricator in Jackson Wyoming?