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I'm not familiar with your door, but I do have a few ideas. My first idea, and the one I'm most optimistic about, is to simply install a surface-mount handle set like our 82-005. These are nice because they have their own self-contained mortise latch, meaning the handle doesn't have to interact with any mechanism in the door. Based on your description, it doesn't sound like your door ever had a mortise latch to begin with, so this seems to me like the best idea.

The 82-005 has a 4-15/16" hole spacing, center-to-center. You should check your existing screw hole spacing to ensure that this matches up. Also note that the 82-005 only latches on the inside. The only handles that latch on the outside are those that use an exterior key lock, but I'm not sure if that's something you want or not. In the case of your door, installing an exterior key lock may be more trouble than it's worth, but let me know if you want to pursue that and we can work from there.

There aren't any patio glass door handles that will do that without the inclusion of a key lock, unfortunately. Like I said, in the case of your door that may be a bit of trouble, but it may be possible to install a key lock if you're willing to drill new holes in the door through which the key can fit, where it will then engage with the interior handle.

If you aren't interested in a key lock, though, then I'm afraid there are no other options. For security purposes, no manufacturer that I know of makes a patio glass door handle set with a latch on the exterior side.