Turn on a multi-functional pressure cooker (such as Instant Pot(R)) and select Saute function. Melt butter and cook onion until soft and tender, about 3 minutes. Add chicken; cook until golden and crispy, about 5 minutes. Place diced tomatoes, kale, chicken broth, salt, garlic powder, oregano, and black pepper in the pot. Close and lock the lid. Select high pressure according to manufacturer's instructions; set timer for 10 minutes. Allow 10 to 15 minutes for pressure to build.
Release pressure using the natural-release method according to manufacturer's instructions, about 10 minutes. Complete releasing pressure carefully using the quick-release method according to manufacturer's instructions, about 5 minutes. Unlock and remove the lid.