A FAQ I receive is how to delete your Instagram account. If you notice, you cannot do it in the app.
Go on your desktop/laptop and go to http://www.instagram.com, you will then go to the “Delete Your Account” page. If you haven’t logged in yet, it will ask you so make sure you know your log-in info.
Instagram will ask you why you are deleting your account and then you will need to re-enter your password.
Once you get through that step, you will be able to permanently delete your account.
Instagram does not have the capability of deleting an account if you can’t log in.
Hope this helps you out!