You have have a difficult set of requirements. Perhaps unobtainable as currently presented.
Having a knife hold a sharp edge for one week is no problem. But it needs to be a high-quality steel to start with, it needs to be properly sharp to start with, and then they need to be nice to it. No dragging the blade across the cutting board or other poor practices.
To address each:
I've seen home cooks scrape along the cutting board, with the knife perpendicular to the board, to gather up cut pieces. Seems like a great way to kill an edge fast.
This was my first thought. Almost everyone in the general public does it, and it’s a simple fix, flip the knife over and use the spine to sweep food.
I’ve seen people say the don’t do this, then I watch them do it without thinking. Some self sabotage without knowing it. If a knife is dulling in a week (with a home cook), they’re probably doing something they shouldn’t.
My next thought was the cutting board, but it sounds like you’ve already gotten/made her a good one.
cheap VG10, Like a used Tojiro DP.
This was my thought as well. However it’s hard enough that it can chip.
It really depends on their knife skills, and how nice they are to it. But above about HRC 58 things will chip if not careful. IIRC Tojiro VG10 is around 60-61.
This is a steel/brand I would recommend for my own mother, but I also wouldn’t be surprised if she chipped it at first while learning to use it. So they will need to be a little careful with it. That said it’s a good balance of durability, ability to hold a sharp edge, and edge retention. As long they’re willing to be nice to it, and accept some possible chips whilst learning.
Otherwise stick to knife below HRC58. Henkel Zwilling has a bolster free chef knife in that range. They regularly go on sale for $60 if you sign up on their website for sale notifications.
The last issue is going to be how it’s sharpened. Or put another way, the quality of the sharpening job.
The sharpener currently being used, to be perfectly honest, is absolutely abysmal. I personally wouldn’t use one, or even recommend one to anyone. They do a very poor job, and chew through steel on a knife like crazy. It’s also possible trying to sharpen a Tojiro VG10 with a sharpener like you’re using could create micro chips in the blade.
If you want to continue to use an electric sharpener, I would recommend a better one. Something like a Tormek, but those are quite expensive. Companies make cheaper clones, or similar lower cost versions, but no idea how well those were work. But any electric sharpener which works halfway decent is going to be expensive ($200-300+).
I would get something better than what you’re currently using. Something that will put a decent edge on the knife, and chew through less steel while doing it. You’re going to have to research here a little, increase your budget, and see what she’s willing to do/learn. But without a better sharpener, even with a decent knife skills and a good knife, you won’t achieve your goals.
Then as shared, having a strop will be helpful. To remove the burr, And help keep it sharp between sharpenings. This will also extend the life of the knife, as it won’t need to sharpened as often, and thus not grind the blade down as fast.
TL,DR:
Make sure she’s practicing good knife skills and not needlessly dulling the blade. Have a decent knife (the Vicorinox is probably fine, but you can certainly get something nicer). Lastly, a knife can’t hold a sharp edge if never had a decent edge to begin with. You’ll probably need a better sharpening system.
Hope this helps.
Cheers,
Nick