Any way to remotely turn on location services without root
No, there is no hope to activate location service remotely without root access or without a system app. Change in the state of location service requires editing secure settings which cannot be done without root access or without system UI (user consent) or without a system app having the permission android.permission.WRITE_SECURE_SETTINGS. Since your Android doesn't meet any criteria I suggest you do not invest your time anymore in this direction.

As for any other method: I do not know but a local law enforcement agency may be able to assist you with common mobile tracking techniques.

Related: Is it possible to enable Location Services via Shell?