Here is the procedure to delete / de-activate your Instagram account permanently.
Step 1: Visit this link, https://www.instagram.com/accounts/login/?next=/accounts/remove/request/permanent/ (Please login if you haven’t already logged in).
Step 2: Click on the drop down button.
Step 3: Select a reason for the deletion of the account. (If there is no reason, you can click on Something else).
Step 4: Enter your password.
Step 5: Click on Delete account.
You will receive a message saying, “Your account will be deleted”.
After you see this message close the tab and don’t try to open your account again as it might get re-activated if you do so.