Can't delete iCloud Account from iPhone 5S

I have an iPhone 5S with the latest ios7 update. I am trying to delete my iCloud account but when I hit the delete button it does nothing. It doesn't prompt me to put in my apple id password. It simply does nothing. I need to delete this account so we can sign in to our new account with our new apple id due to an email address change. Have tried everything. All other apple devices have worked just fine. This one just does nothing. Can anyone help?