Brown the sausage in a skillet over medium heat, breaking it into small pieces as it cooks. Drain excess fat and transfer the sausage to the slow cooker.
Add the diced potatoes, minced garlic, diced onion, chicken broth, salt, and black pepper to the slow cooker. Stir to combine.
Cover and cook on low for 6 hours or until the potatoes are very soft.
Use an immersion blender to partially blend the soup, leaving some potato chunks for texture. If you do not have an immersion blender, mash some of the potatoes against the side of the slow cooker with a spoon.
Stir in the chopped kale and heavy cream. Cover and cook on low for 20 more minutes until the kale is wilted and tender.