So do you feel that a mother who leaves a baby on a playmat on the floor for ten minutes so she can fold laundry is somehow shirking her duty? Or that she is treating the child as having low value? My kids have always loved to cuddle as babies/toddlers but also been happy for windows of time when I could get things done, or been content in a carrier for a bit, or napped. My kids would have been cranky as all get out had I wanted to interact with them 24/7. Babies like some chill-out time, too. Not for hours and hours a day, but 15 minutes here or there while I heat a can of soup, or put some laundry away, or wipe down a counter. Especially with toddlers, who can be taught to "help" with minor tasks or encouraged to play kitchen while you do kitchen tasks, etc.
Just as you wouldn't work for someone who expected light housekeeping, I wouldn't hire someone who refused any cleaning and acted so disrespectful. If many moms like me can juggle raising multiple children, keeping their homes decently picked up (not white glove clean, just basic tidiness/decluttered), being wives, shuttling kids around, cooking meals, etc, then I should expect a full time nanny who is paid decently to expect to do light work associated with the kid(s) in her downtime.
I also wanted to add that even in daycare centers, some work is expected. I know the teachers in my son's school (he attends a SN preschool with an infant center attached to it) are responsible for heating and serving meals, cleaning up from them, sweeping the floor and sanitizing the tables/highchairs after each meal, taking out diaper trash, cleaning up toys after each activity, wiping down counters, etc. Multiple times per day. And there are at least 3:1 ratios in those infant rooms. I know the people there don't get paid much, even. So it's apparently not totally out of the realm of expectation or ability.
As a mom, I have no trouble doing that kind of thing (although I don't do it every day or every hour or whatever...it varies. Some days I forge waaaaay ahead on house/life projects and some days I don't).
But if I were a nanny I think I would be more focused/cautious about not minding my charges. A lot would depend on the environment too. Is the kitchen right next to an area that they can play safely and so on.
When I babysit other people's kids, I find it harder than dealing with my own...we don't have the same connection, the same rhythm; my sense of where they are and what they are doing is not as in tune. I have to work a lot harder to watch my friend's son for an hour than to watch mine. I have a friend with twins and honest - they have an amazing capacity to tag-team dangerous situations. I'm a bit edgy with them.
It also doesn't sound to me like the nanny is trashing everything or not doing any housework at all - she does shopping and presumably puts that away, some laundry, some tidying up, etc. It's just that not every single thing is done every single day. Also, although staff clean at daycares they also have breaks. They have backup if they have to pee or if they are about to lose it. The trouble with always being busy tidying every last thing up is that you can end up with no reserves at a moment when you really want them. I have a 9 week old baby and a 5 year old and my husband was away for a weekend for work recently (when my newborn was 7 weeks old). Saturday I was very productive only to find myself losing patience Sunday afternoon. Next time fewer chores will get done, even if I end up with energy leftover.
When I've had a nanny I've always wanted to be sure that she knew it was all right with me if she needed to take nap time to recover. It's way more important to me that my child's caregiver be grounded and even happy than to get every penny's worth...because I want a caregiver who is not punching a clock but being creative, positive and so on.
Anyways I'm sure the OP will work it out - she seems pretty set on her course. OP I came back because I had another thought - what if you and your nanny agreed she would stay 15 min later to take care of these things (basically to pretty the house up for you), and then she would get an extra day's vacation pay every quarter or something like that? It might be a very small financial investment to make your life better, and still leave the reasonable idea that she is doing her best already and for whatever reason those things are the tipping point for her in a day.