That example involves at least three sprites that I can see: the player's head and arm are two sprites that always look at the mouse, and the player's body does not look at anything but flips when the mouse is to the left of the player.
You can probably do this without checking the angle; just check if the mouse coordinate is to the left of the player, assuming the player sprite faces right by default. If you were to code it this way it might look something like this (I did not run this code):
extends Sprite
func _process(delta):
var vert = get_global_mouse_position()
var gpos = self.get_global_position()
if gpos.x < vert.x:
self.set_flip_h(true)
vert.x = -vert.x # our sprite would be facing away from the mouse after flipping
else:
self.set_flip_h(false)
look_at(vert)
Note that to get it to look like that video you'll have to have three sprites in your player scene.
The arm and head should be rotated and flipped; you'll have to either add or subtract 180 degrees when the sprite is flipped.
The body should not be rotated, just flipped. The code above works with one sprite like the head or arm. I don't know if I got the rotation handled correctly.
EDIT: fixed an issue with the y coordinate direction getting inverted