Deleting your Twitch account is easy. Twitch offers a disable your account option but the outcome is same as having your account deleted. Follow the steps below to delete your Twitch account.
Delete Twitch - Website - Slides & Instructions
Deleting Twitch? Share Your Experiences
Questions & Answers
Twitch offers a disable option which is the closest you can get to account deletion. After disabling your account you will no longer be able to login and you will loose all of your saved settings. However, it is not clear what happens to your personal details. At least your username seems to be stored forever, because you cannot reuse it to create a new account, even after a few years!