OP, I read some of your thread yesterday but didn't post, sorry. I am sorry to hear of your son's illness and the pressures you are under. I have one child (DS, who is nearly 4 and thankfully is very well) and work part time (3 long days a week) in a professional role, and we have a nanny too so I can empathise to a degree. Not in relation to the illness, sorry, you have my utmost sympathy there.
Nothing you have thought of asking your nanny to do is unreasonable. Your nanny isn't doing the basic duties that are in her contract at the moment, which is one issue. Another is that you are clearly under enormous pressure, which she must be aware of, working in your home, and not only is she not doing her basic job, but she somehow hasn't had the decency to offer to help out more as it is clearly needed.
Is she your first nanny by any chance? I ask as we are on our second nanny. The first nanny stayed with us for 18 months before taking maternity leave. The second (current) nanny has been with us for 18 months too and hopefully isn't going anywhere. They have both been terrific with our DS, and I have no complaints in the level of care or love he has received from either of them.
But the difference that our current nanny makes to our lives as a family is enormous. She has so much energy and initiative - she will put a load of washing on, peg it on the line, bring it in, iron the parts that are for DS and either fold and leave our things on our bed, or put them in our ironing pile. She will make a cake every week (with DS as he loves to cook), which is a lovely treat for us. She gets supplies like bread and milk in if we're running low, puts food from the fridge into the freezer if its on its use by date, always leaves the playroom clean and tidy at the end of the day, and just generally sees things that need to be done and organises her time around them. All whilst caring for DS and doing lovely things with him. Every day.
Sorry for the essay, but I know first hand the difference between a fairly good nanny and an excellent nanny. When I found out that our first nanny was going on maternity leave, I was very fretful about how DS would cope being cared for by someone else, how we would all get on with a new person etc, but this has been such a positive change for us. Would you consider recruiting a new nanny? It could make all the difference for you. I appreciate that it will also take time and effort, and you may not have the energy to do that now, but it could be something to think about as your son improves from this latest bout of illness.
In the emantime, I think you need to be more direct with your nanny, and spell out what needs to be done. maybe give her a timetable (in the loosest sense of the word) with what you expect to be done. She may try to resist it, but maybe that would help make the decision on whether she is right or wrong for you as a family long term a little easier. She may enjoy the challenge and reward.