After a lot of trial and error, including trying a whole new HC-05 module I have come up with the conclusion that you can't use the Keyboard.h library with the HC-05 bluetooth chip.
I think it's something to do with the built-in USB communication chip on the arduino micro which the Keyboard.h library uses to identify to the PC that it is a keyboard/HID.
I tried changing the HC-05 class to a keyboard (AT+CLASS=000540) which managed to change the symbol of the bluetooth device on the PC to a wee keyboard but it still didn't come up in the "Mouse, keyboard & pen" field of the "bluetooth and other devices" panel. Compared to when you plug the arduino micro directly to the PC using a USB cable it would appear in the "Mouse, keyboard & pen" field as a keyboard (with the Keyboard.h library loaded to the arduino)
I'm sure there are ways to make a bluetooth keyboard using an arduino but it is not by using the Keyboard.h library.
What other ways are there of sending keyboard presses to the PC?
Can the HC-05 definitely be used to send keyboard/HID commands to the PC? Or do I need to look into an RN-42 module?