How can I turn Bluetooth on when I don’t have a mouse

What year and model is your Mac?

What keyboard and mouse do you have?

If you have a Magic Keyboard, plug it into the Mac using the charging cable and follow dialabrain's instruction.

If you have the older keyboard and mouse with replaceable batteries? Then you need to dig out, barrow or buy an inexpensive USB keyboard and mouse, so that you can turn on the bluetooth and repair your wireless devices.

Assuming you have a working keyboard, press Command+Space to open Spotlight, type in bluetooth, scroll to "Bluetooth File Exchange" using the arrow keys, press return.