The weight thing is a mistake and if you focus on that you'll lose credibility and ethical stand point.
I'm going to ignore this.
Do you see this nanny workkmg out?
What would she have to do to regain your trust and put your concerns to rest? It's not her role to do this btw, it's just a way to understand whether there is any way this employee/ employer relationship can progress.
Or do you just want to say it's not worked out? and rapidly get on finding someone who is a better fit, and let this nanny find somewhere right for her too.
There doesn't have to be a goodie and a baddie in this situation, and don't feel like you have to keep on trying if it's not right.
Attendance and illness are tricky things. Even without all the ethical debates around it, you can't be discriminating against someone because of things outside their control. But you also need someone who will be able to keep up with the physical demands of the job and you are still in the probation period where you can make this kind of decision.
So much illness so early on in a new job, may be just really bad luck, or a sign of things to come with continuing ill health, or actually a sign that she's not happy, or not ready for a job which needs a high level of reliability and commitment.
Can you work out which it is?
Last year I made a mistake with a nanny who kept being poorly or needing compassionate leave for one emergency after another. I was a bit of a mug and kept believing this time was the last time and having lots of sympathy. I paid the nanny full wages which was beyond my obligations, and as it was a different thing each week it made me carry on for 3 months. In which she'd only worked 1/2 the time but had been paid full time plus all the money I spent on emergency childcare and days off work.
I then was advised by the agency that they had seen this happen before and it was more likely the nanny just didn't want the job, and that I should always bear in mind what absence can be communicating. I was shocked as I bent over backwards for that nanny and yes, she left and didn't turn up for her notice and sent angry emails demanding payment for her notice period even though it wasn't worked!
Anyway, my point is that as an employer you have to Err on the side of caution and be very careful what you say as you have a responsibility and a duty towards your employee, but you don't have to be a push over either.
Good luck whatever you decide