Reset Security Questions

Do not try to sign in or change your security questions for a full 24 hours. After 24 hours use a device and Wi-Fi you’ve used to sign into the Apple ID with before and try resetting your security questions one more time.

If this still doesn’t work your next option is to try to enable to factor authentication following Two-factor authentication for Apple ID - Apple Support

For the security questions, if you are being told there isn't enough information to reset them, contact Apple ID Support, although based on multiple posts I have read, they will not help you recover the answers to your security questions.

From the article below. 

Without proof of your identity via security questions and other carefully selected criteria, Apple Support can't help you reset a password or perform any other actions on your account. These policies are audited and reviewed on a regular basis.

Security and your Apple ID.    *

Apple ID - Official Apple Support.    -

Contact Apple for help with Apple ID account security - Apple Support.  * (Link doesn't always load)  

Get Support   - Work through the screens to get a chat option. 

Apple Support schedule phone call. -

Apple on Twitter

Messages chat with Apple

➢ Change Apple ID security questions.

To reset security questions:

➢ Contact Apple for help with Apple ID account security. 