You can't delete the existing account without the password. If the old ID is yours, and if your new ID was created by editing the details of this old ID (rather than being an entirely new ID), go to https://appleid.apple.com, click Manage my Apple ID and sign in with your current iCloud ID. Click edit next to the primary email account, change it back to your old email address and save the change. Then edit the name of the account to change it back to your old email address. You can now use your current password to turn off Find My iDevice, even though it prompts you for the password for your old account ID. Then save any photo stream photos that you wish to keep to your camera roll. When finished go to Settings>iCloud, tap Delete Account and choose Delete from My iDevice when prompted (your iCloud data will still be in iCloud). Next, go back to https://appleid.apple.com and change your primary email address and iCloud ID name back to the way it was. Now you can go to Settings>iCloud and sign in with your current iCloud ID and password.