Ok, this is a very geometric argument but using simple geometry so don't get too freaked out.. 

Here is my initial diagram.. With the face and the mirror 

ABCD make up my face.. To simplify things.. I had to assume that the distance from one side of the face to the closest eye is the same on both sides.. This means that AB = CD

The line containing EF make up my mirror.

 The mirror and the face are parallel to each other. 

Now, light rights from the edge of my face A and D are going to shoot towards the mirror..



For simplifcation, I only showed light rays coming from the side of the face fartherest from each eye. These light rays are the ones that really matters. If they can reach the eyes, then both of your eyes can see both side of your face.  (This is because the side of the face closer to each eye is already close enough to the eye for the eyes to see them, what really matteris is  if each eye can see the side of the face that is fartherest from   them) These light rays are going to hit the edge of the mirror.

Now I am going to draw a straight line from the point where these light rays hit and reflect off the mirror (point E and F) to the face. Theses are perpendicular to both the face and the mirror (since both face and mirror are parallel to each other)

Now, we know the angle of incidence = angle of reflection.. But the thing is.. the angle of incidence and angle of reflection on both sides are the same.. This is due to the fact that both eyes see the same amount of the face fartherest from them.. (This part is hard to see and hard to explain but think the mirror is located right at the center, at the same distance from both edge of the face and it may make more sense..)

Yeah as you can see.. a, b represents angles. a+ b = 90 degrees (again, since these lines are perpendicular to both face and mirror) So if you play around you can see that EFAB makes up a parallogram and EFCD makes up another parallogram. These two parallograms are the same since they have the same side EF and same angles etc.. Now, if we can find out how long AB + CD is, then we can divide that result by two to get AB or CD since AB = CD. 

To figure out how long AB+CD is, we use the length of the face - distance between two eyes. Now, we divide that by 2 to get length of AB (or CD). Since it is is a parallelogram, we can easily see that AB = EF so that's the lenght of our mirror.

I hope that made sense.. is harder to explain than I thought but if you play with the geometry abit it should make sense..