The Instagram app doesn't have a delete account option like the Facebook app has. So here's the trick.
Follow along in the video and you will exactly find how you can delete your Instagram account or you can even disable your account.
1. Open your app
2. Tap on the 3 dots on the top right corner
3. Click on Help, then Help center
This will prompt you to open a browser
4. Click on manage account
5. Click on delete account