As a Nanny caring for 3 children, 2 of whom are under 5, not a Nanny Housekeeper - I would do:
maintaining the children's areas - make the children's beds
general laundry, including children's bedding (no ironing)
cooking their meals
I sweep and vac on occasion
if the dishwasher needs emptying I will empty it. I will also fill it, put it on, of leave for parents to complete filling, depending on how much is in it.
So given that those I see as being all fairly usual nanny duties for a nanny who is caring for children all day... if all your children are at school, adding on some duties on top of that such as cleaning the bathroom, dusting, ironing, doe not sound unreasonable to me.
As Frackel says, if the job description was clear enough then she is not fulfilling that job description. She is in probation period, so either bucks her idea up or she leaves.
Doing some of the tasks with her own 1 year old in tow could well present problems. However if things are timed right, then I would still expect all those things to get done over the course of a week.
Why does she not have set hours of work? Or does she have set hours? I would suggest you tell her what time she is to arrive... not leave it up to her. Say something along the lines of "I'm working from home a bit in the morning, is it possible for you to come in at 10am, rather than 8am?" Also consider if it's even sensible to do that... even if you are doing the school run, she has domestic duties to do so it's not a problem her being there at usual time is it? I realise that you may have been trying to be nice but it messes with routine. She may be better at doing things routinely, as may her child.