Did I cause my sister's illness by praying against my mother, and how do I repent from my feeling of guilt?

You bear no sin for praying that your mother's harm be averted. Ibn Baz was asked about supplicating against an unjust father and he said: "It is not permissible for you to supplicate against him, but you should say: 'O Allah, guide him; O Allah, protect us from his evil; Allah is sufficient for us, and [He is] the best Disposer of affairs.' There is no harm in this. But supplicating against him, no..." What befell your sister was by divine decree, so do not blame yourself. Increase your prayers for your sister's well-being and for your mother's guidance.