there is a stupendously huge gap, so a steel box with a mortice lock would not work if mounted in the usual way, flush with the stile.
I would be thinking about a steel-cased rimlock mounted on the inner face of the gate, probably bolted to a custom-made plate welded or clamped to the gate. Your problem will be that because of the huge gap, the lock and/or staple will have to protrude so they meet, and you will catch the protruding lock every time you walk past, unless you put a piece of steel angle on the gate or the fence, for the gate to shut against.
the sort of lock made for sheds is useless, but there are some good ones branded Wellington or Imperial that are used on utility cabins. I'd get one with a knob-operated latch as well as the lock. You can also get FB rimlocks with a standard key that the fire brigade carry.
Is it for your front door, and frequently used?