Good grief Dizzy you certainly are overly invested in this, aren't you? 
OP - if she can't get a job due to your hours of work, how can you reasonably nitpick that she isn't working while you are? Do you really not see how that doesn't make sense?
I do think you need to have a conversation with her - but not along the lines of "what the flaming heck do you do all day?" because that's a fast track to nowhere. And it's insulting.
Being a parent is something you BOTH should be doing - if you are both home, you are BOTH parents - hence sharing the tasks.
But she is obviously upset about something specific in the morning routine, so perhaps a non-confrontational discussion about it would be more helpful.
I highly doubt that she literally does nothing all day while you're working. I don't work, have 2 children, and I spend my time cleaning, baking, cooking, running errands, taking the dcs to various appointments, doing odd things here and there like (for example) taking the car in to get a tyre replaced tomorrow and then to the other car place to get the brakes checked. These kind of little things take time. Then there's laundry, washing up. School runs in the afternoon. And then when the kids are home, it's helping with homework, monitoring them, breaking up arguments, organising a snack, getting tea ready, tidying up, baths, bedtime schedules, etc etc
So her work schedule may not be 9-5, but it starts early and goes late, even if she or the kids are ill, every day even on weekends (extra time because on weekends they're home all day). So I think you need to re-evaluate why you think she has the day to herself - because I'll bet she has a list of things to do a mile long that you're not even aware of.