Can't delete old iCloud backup

Question

Had a phone set up to back up to iCloud. That phone went dead, and was returned to Apple and exchanged with a new one. Because the screen went dead -- thus rendering the phone inoperable -- I was unable to shut off iCloud backups and delete the old backup.
Right now, I can't delete the old phone's iCloud backup, and I don't have enough space to start backups on the replacement phone. How can I resolve this?

Answer 1

If you registered your old phone with Apple, try unregistering it by:
Going to: https://supportprofile.apple.com/MySupportProfile.do
Loging in if not already logged in.
Clicking on ">" to the right of the phone listing
Clicking "unregister"
This may take a day or two but hopefully will then allow you to remove the backup. 
If it doesn't, then one option is to create a new iCloud account and migrate your data to the new account.  To do this, go to Settings>iCloud and turn all data that is syncing with iCloud (contacts, calendars, etc.) to Off.  When prompted choose to keep the data on the iPhone.  After everything is turned off, scroll to the bottom and tap Delete Account.  (This will only delete the account from this phone, not from iCloud, in case you ever want to re-use the account in the future.)  Next, set up a new iCloud account on this phone using a different Apple ID and turn iCloud data syncing for contacts, etc. back to On.  When prompted, choose Merge.  This will upload the data to this new account.