Let the potatoes simmer in the broth while you prep and cook the remaining ingredients. Because the kielbasa is quite salty, no added salt is necessary.
Bring broth, potatoes, and bay leaf to boil in large saucepan over medium-high heat. Reduce heat to medium-low and simmer until potatoes are tender, about 10 minutes. Discard bay leaf. Using potato masher, coarsely break up potatoes, leaving some large chunks.
Meanwhile, melt butter in Dutch oven over medium heat. Cook kielbasa, stirring frequently, until lightly browned in spots, about 4 minutes. Add leeks and cook until soft, about 4 minutes.
Add potato mixture and kale to Dutch oven and simmer until kale is tender, about 5 minutes. Season with pepper and serve.