Prepare your pressure canner, jars, and lids. Add a few inches of water to the canner and keep the jars hot at about 180 degrees F.
Peel and cube the potatoes into even 1/2-inch cubes. Destem and chop the kale, using mostly the leaves.
Brown the sausage in a large stockpot over medium heat until no pink remains, leaving some small chunks rather than fine crumbles.
Add the potatoes, kale, salt, pepper, and broth or water to the pot. Bring to a simmer and cook until heated through, about 5 minutes.
Remove from heat. Ladle the solids evenly into the hot jars first, then divide the broth among the jars, leaving 1 inch headspace.
Remove air bubbles, recheck the headspace, and top off with boiling broth or water if needed. Wipe the rims, then apply lids and bands fingertip tight.
Place the jars in the pressure canner and lock the lid. Vent steam for 10 minutes, then bring the canner to pressure. Process pints for 75 minutes or quarts for 90 minutes at 10 lbs pressure in a weighted gauge canner or 11 lbs pressure in a dial gauge canner, adjusting pressure for altitude.
When the processing time is complete, turn off the heat and allow the canner to depressurize naturally. Do not force cool. Once the canner reaches zero pressure, remove the weight or open the petcock and wait 10 minutes before opening the lid.
Remove the jars and cool on the counter for at least 12 hours. Check the seals and store sealed jars; refrigerate any that did not seal.