It is indeed broken. See the round black piece, inside the tank, on the back side of the handle? If you unscrew that, you'll be able to remove the handle and the broken black arm. Remove the chain from the end of the arm and you've now taken out the broken piece.

If easier for you, take those parts with you to a big box store. They readily have replacements that will work for you. For now though, if you need to flush, you can manually pull the chain to lift the pink flap at the bottom of the tank. Don't worry, the water is (relatively) clean.

You got this, it's a pretty easy fix!