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In my experience grass is great at breaking down, better than paper and veg waste. Just have to mix it in a bit and keep it wet.
Things I've found helpful are: water, worms (Eisenia fetida or Dendrobaena, L.terrestris -big earthworms - aren't a compost worm), ants and heat. Two you can add, one you can encourage and the other you'll have to wait for. You can buy worms but if you leave a few planks on the ground for a week or two, the reddy coloured worms that appear under them will be best for composting. They'll reproduce easily and you'll soon have tons of them.
It sounds like it's full of stuff that is notoriously slow to compost, i.e. grass, twigs, etc. Are you adding any vegetable waste? Possibly add torn up paper too, just to accelerate things?