Q: How can I switch off 'Find my phone' if I have lost my password for an old Apple ID that no longer exists, but my phone still recognises? I cannot delete the ID either because I need the password. Any ideas?

Have you tried resetting the password for that particular Apple ID: If you forgot your Apple ID password - Apple Support

Apple IDs, never cease to exist. While the associated E-Mail address may no longer function, the Apple ID will continue to work regardless.

If you don't remember the Password, you can try to recover it as suggested above.

Otherwise you will need to contact iTunes Support to have them reset the password for you.

Contact Apple for help with Apple ID account security - Apple Support

Once you recover the password, or have it reset,  I suggest you go update the associated E-Mail address.