What is the expiation for accidentally killing a cat?
The Dharma Shastras prescribe some expiations for unintentionally killing a cat.
From the Vishnu Smriti:
If he has unintentionally killed a mouse, or a cat, or an ichneumon, or a frog, or a Duṇḍubha snake, or a large serpent—he must fast for one day, and on the next day give a dish of milk, sesamum and rice mixed together to a Brāhmaṇa and give him an iron hoe as his fee.
From the Yājñavalkya Smriti:
(3.271) For killing a cat, an alligator, an ichneumon, a frog or birds, one should drink milk for three days, or perform a quarter of the Kṛcchra penance.
The Vishnu Smriti verse above says explicitly that that expiation is for the unintentional killing of a cat, so you should follow that.
Just go to a nearby temple and present a Purohit those two gifts mentioned in that verse.