OH was at the allotment doing exactly the same this afternoon.
Sprout stalks were put in compost bin ( large green spaceship style) and chopped up into 3-4 inch chunks with a sharp spade. Later in the year the contents of this bin will get transferred into the smaller one ( normal sized black one) to finish composting down.
What animals do you have?
Goats, and I am assuming sheep, will eat the stalks if they are split down the middle. All that may be left will be chewed outer ?skin?. Keeps them occupied and stops them munching on sheds
Perhaps poultry will enjoy the split stalks hung up?
I am sure someone will be along soon and help with the Kale question