I see that you posted under a carpentry tag. however, my solution would be to find a section of Aluminium Tubing, or tubing profile which fits the "gap" which you describe.
As You can see, the example picture is a rectangular profile, which will run the entire height of the door, To fill the specified gap.
Your problem will be to attach said profile to the side of the slider. essentially the best thing would be to drill into the frame and insert pop rivets, however, there will be a glass pane on the other side of the aluminium door frame, so drilling will be difficult.
The profile below will allow you to drill your rivets on the side of the door (ie the narrow face of the sliding profile, then you can drill two holes on the top and bottom of the slider (in your photo its the side you can see.
You will then be able to mount your lock flush to the hinge door.
Above you can see what I mean. I must stress here, that you dont want to break the glass pane by drilling into the frame where there is glass.
Answering my own question, but I did eventually get the doors to behave themselves. The way I ended up doing it was this: The rollers that I got with the hardware kit and adjustable. They have one fixed screw hole and two more that are curved slots letting you not adjust the angle. I got the door in by removing the rollers from the door, placing the rollers in the track and then screwing them into the door while they were hanging from the rails. They ended up very slightly angled, but it looks fine and, more importantly, they can open and close. I had to adjust them again slightly because the door was dragging on the floor guide at the end of it's run.
It wasn't easy - an assistant would have been a great help. But I did manage to get them in and have them hanging in a way that I could move them without them falling off again. I used some shims to lift up the bottom of the door while I screwed it back in since it would be impossible to hold the door off the floor and turn the screws at the same time. Again, an assistant would have made it easier.
Best Answer
Even though someone is using it as a closet, that is an "alcove", so it was not built to have a door. So you will have fitment issue no matter what you do; the crown molding at the top is a problem, the baseboard on the left side it another, where a sliding door would slide to is a problem, etc. etc. etc.
Were it me, I would make a new top header extending down just below that crown molding and add some right side trim to give a straight edge for it to close against, then use a "barn door" external track system on the wall to the right. Ignore the color and style of this door, but here is an example of the type of slide system.