Go to the website by clicking here and Instagram in the bottom of the site, select 'Your Account' (can not be done from your phone!).
In the 'Edit Profile', go to the bottom and click 'I would delete my account' again.
Choose from the drop down menu the reason you want to delete your profile on Instagram and Ready!
Delete the application from your smartphone; If you do not know yet how to do it, you can learn with this tutorial.