Try to sign in to your @live.com email address. If you get a "doesn't exist" error message, this means that the alias has already been deleted/removed.
Unfortunately, Microsoft has a policy that once a @live.com email alias is removed, it is permanently deleted and cannot be added back or used to create a new account because Microsoft does not recycle email addresses ending in @live.com.
If this is associated with your Apple ID, try contacting Apple Support to see if they have another option to help you recover your account with them.