How can I restore an iCloud back-up to a new device when old device no longer works?
According to this Apple Support page:

When you have two-step verification for your Apple ID, you'll always need at least two of the following to sign in:


Your Apple ID password
Access to one of your trusted devices
Your Recovery Key


If you've permanently lost any two of these items, you can't sign in or regain access to your account. You'll need to create a new Apple ID. You can create a new one from the Apple ID site.