Let me start off by saying that I don't know the answers, but I will throw some stuff out there anyway. Disclaimer: I am not looking at the manual and just looking at my radio.
When you hit the menu button, at the bottom of the first menu screen, there is a GPS tab. Clicking on it gives you two options: GPS TX Mode, and GPS Auto TX.
When you click on the tab, GPS TX Mode, you need to have this turned on to send your GPS data; in this case D-PRS (DV-A). If this isn't turned on, you won't show up in APRS.fi
The other tab, GPS Auto TX does exactly what you think it would do: it automatically transmits your location information every so often depending on what time you set it for: every five minutes, 10 minutes, or every 30 minutes. This doesn't sound like it effects what you are asking about.
So, on the XYL's radio make sure the GPS TX Mode is on and set to D-PRS.
Another possible issue is that the the radio can't send GPS data if the internal GPS is not locked. In other words, when you first turn it on, or maybe if you are in the basement or something the GPS isn't exchanging information with the satellites so it can't send your location data. You need to give it a little bit of time to aquire the GPS satellites. On this radio, this happens fairly quickly; maybe a minute if you are standing outside. The internal GPS also seems to work well and for me seems to "find" the GPS satellites even when I am in my basement. When that little satellite icon at the top of the screen (in the center) stops flashing, you have satellite lock and a valid location fix.
I doubt that this helped you, but maybe you can confirm that this stuff is done and we can go from there. I am not sure I can help you find the solution if it is something other than these few things I already mentioned, but I wanted to let you know someone was reading the post. If you are using a repeater, obviously the repeater has to pass this data on to the internet. I don't know anything about D-Star repeaters and how they work but it seems obvious that this has to happen. What happens behind the scenes with these digital voice modes is something I don't know a lot about and even if I did, I am not sure it is possible to know exactly what happened when a problem occurs. It's a complicated system. You have the basic radio parts: am I in a good enough location with a good enough antenna and enough RF power to be getting into the repeater ? And then you also have the repeater itself. But beyond that you have all kinds of data/internet things going on that you don't have any control over and no way to know what is going on with that.