Any way to remotely turn on location services without root so I lost my phone this morning and I tried to use the standard android device manager to track it but it's useless without location services, which I turn off due to it draining too much battery power.

Is there any way to turn it on so I can used ADM or is there another method which can track my phone without location services, maybe using something like wifi hotspots?

I'm not sure which exact android firmware I have but it's above 5.0.0 if that's of any help and it's a xperia z1
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?