I don't think that a handy box is rated for a recessed installation and why would you use ac90 in a house? Use an 1104 or 1004 with ears and clamps and nmd90. It's better to attach to a stud, but every one seems to be using F clips. The flex bit is good for getting you out of a problem, but it's also great for getting you into one, so be careful. If you can use a flex bit, you must have access above or below so an auger bit will work, too.
The hole for the box is 2.25 inches wide by 3.25 inches high and the side are straight. Notch the bottom for the screw tab so the box can sit on the bottom of the hole. If you can hit a stud, use #6 wood screws. They sit tighter against the side of the box, but the hole must be .25 inches away from the stud or the box will twist out of the wall.
I assume it isn't an outside wall or you wouldn't be thinking about using a mirror; but, if it is, don't forget the vapour barrier.
If the walls are lathe and plaster, just call an electrician.