... You need to hang it on something. A 20" wide mirror is a good amount of weight to be hanging in drywall.
If I was doing the job, I'd either hang it where there IS a stud (so it's a bit off-center), or, I'd cut out the wall a little, 'cripple' a horrizontal 2x4 into the wall cavity (and put the wall back and all that), and THEN nail into the cripple. I think you'd be taking a more-than-minor risk by hanging it right on the drywall. Talk to the Hardware folks at the Home Center. See if they have an idea.