You might be surprised how strong a nicely fitting miter joint can be with no reinforcement.
I recently made this with no reinforcements. The wood is only 3/8" thick by 3/4" wide and I used Titebond III
This was a prototype and I wanted to test how strong the joints would be, so I tried to crush it in my hand.
It took more strength than I expected and some of the joints could only be destroyed using a clamp. Needless to say, I was surprised.
A couple of years ago, I made some larger picture frames for our church and put splines in the mitered joints. I can't imagine the joints ever coming apart. If you put splines in the joints of your mirror, I think it will be plenty strong. If you're still unsure, use two splines in each joint.
That said, if you rabbet the back of the inside part of the frame and use a piece of plywood to hold the mirror in place, that will add strength, even using a few screws.
I tend to over-engineer a lot of things. I'd consider glue in the joint and a spline to be plenty, even for a piece as big as you're planning to make.
Blaine