You have a height h. What is the minimum height possible for a plane mirror so that you can see your entire reflection in it without moving your head?
No matter how far you are from the mirror, the minimum height necessary is h/2. The mirror has to be positioned just right, though.
At what height should the very top of the mirror be placed?
The top of the mirror should be at a height halfway between the top of your head and eye level. That's the point on the mirror a ray of light from the top of your head reflects off on its way to entering your eye. Similarly, the bottom of the mirror should be at a height equal to half the distance between your eyes and the floor.