You could first make a Vector2 pointing from the sprite to the mouse:

var vectorToTarget:Vector2 = FlxG.mouse.getPosition() - this.getPosition(); // subtracting the target position vector from the entity's position vector gives us a vector pointing from us to the target

then, in the update() method, set the sprite's angle to the angle of the vector:

this.angle = MathUtil.radToDeg(vectorToTarget.angle); // the Vector2's angle is in radians so we need to convert it to degrees

and check if the vector is pointing left or right, flipping the sprite accordingly:

if(vectorToTarget.x >= 0) {
this.flipY = false;
} else {
this.flipY = true;
}

You could also decide to use a FlxVector instead of a Vector2, in that case you'll need to use the method .subtract() instead of just doing a subtraction operation when creating the vector, as the FlxVector class doesn't overload the operators.

You also won't need to convert radians to degrees as FlxVector measure their angle in degrees already in FlxVector.degrees.

Last thing to note:

I'm sure the code I've written here will need tweaking for your usecase, treat it as just a starting point, reply if you need more pointers.