It looks like sloppy assembly on the insert. Notice how the top rail extends past the side (stile).

Also, something isn't plumb. Either the whole patio door is out of plumb or just the insert. With the insert removed, does the door close properly with no gap?

Attach foam weatherstrip tape to the dog door side.

foam tape

I would try to trim the top rail of the insert back to where it is flush with side stile of the insert. Extend a straight line from the edge of the stile up across the rail. Assuming the insert is aluminum or vinyl, use a file and carefully remove material to the line. That will help close the gap somewhat. Then use a self adhesive weatherstrip as Graywings^^ suggested.

Actually, I would uninstall the insert & reinstall it, shimming where necessary, to try to get it plumb. (or out of plumb, whichever it takes to close the gap on the slider door.) Uninstalled, it will be easier to file that excess off the top rail, too. Then add the weatherstripping if it is still needed.

Those door types are unwieldy at best, even when installed correctly. They never seal well, and tend to break often. A better choice is through the wall door, that leads out into a protected overhang.

The other problem with such inserts is that the gap they leave between the two panes of glass in the partly-open door admits un-conditioned air and insects. Air conditioners in double-hung and slider windows have the same problem. The fix for both is to stuff closed-cell foam into the gaps.

DavidR can you possibly share a link or photo of the type of closed cell foam you think could work for this gap? We will of course need it to stick to one side and not the other, so we can still actually use the door.

Live_wire-oak what you are suggesting is not an option for us.

I have the same issue. Go to ace and get some weatherstripping, I used rubber, but I know exactly what is gonna happen when you put it on the door- the door will still have a gap because the weatherstripping is equal all the way down the door. What id suggest is cutting a wooden dowel, like you would use for security and locking the ajar sliding door, but cut it to be about a quarter inch too long. That way, when you compress the door against the weather stripping, it will compress the area without a gap and the stripping will fill the gap nicely.

I'm having the same issue. I purchased a 3rd door like the one shown because the other two were garbage. The new door is sturdy!! I did put the wooden dowel in as a lock, and I do notice it compressing the two doors, but it's still not enough. I'm thinking of running sheet rock tape down between the two doors to form a temporary shield. I know I will have to take it off every time we open the door, but it's the best I can think of for now. This is with the wooden dowel in. I even try to shimmy the pet door to the right a little and lock it in that way.