Place the chopped kale in a pot with water. Bring to a boil, then reduce the heat to low. Cover and simmer until the kale is tender, about 8-12 minutes. Drain, but reserve the cooking water. Set aside.
Heat half the olive oil in a pot over medium heat. Stir in the onions and cook until just beginning to brown, about 8 minutes.
Stir in the garlic and cook for 1 minute.
Add the cooked kale, the remainder of the olive oil, and the reserved cooking water.
Simmer, uncovered, over medium-high heat until the liquid has nearly evaporated, 8-12 minutes.
Add the green pepper, lemon juice, salt, turmeric, paprika, allspice, and ginger.
Cook until the peppers are soft, about 3-5 minutes. Add salt (according to taste), and cook for an extra 2 minutes.
Serve with injera.