Not sure about the legitimacy/legalities of your particular arrangement, but honestly it sounds very messy and, if the trust has gone, I’d be looking to end the arrangement rather than formalise it further.
You can’t necessarily just agree between yourselves that she is self-employed – employment/tax status depends on the reality of the working arrangement. If HMRC considers her an employee, you could potentially be liable for PAYE/NI and possibly interest or penalties, so I’d get proper advice on that fairly urgently rather than continuing as you are.
I’m also slightly confused by the Nannytax point. Are you already subscribed to them? We use them for our nanny and it’s a paid payroll/HR service, with the HR advice included as part of the subscription, are you sure they're going to advise you if you’re not actually a client?
On the sick pay, anything beyond statutory entitlement is discretionary in our nanny’s arrangement; there isn’t an automatic entitlement to almost full wages just because she was off sick. And if she told you she was too unwell to nanny for you but was actually working full-time doing the same job for another family on those days, I’d be asking her to explain that before paying anything discretionary. Depending on the facts, dishonesty around sickness absence could potentially become a serious disciplinary issue.
Personally I’d stop trying to patch this arrangement up and get proper payroll/employment advice now, establish what her actual employment status has been, sort out anything that needs correcting with HMRC, and then end the arrangement properly if that’s what you decide. Continuing informally when you already know there may be an employment-status issue could leave you in a worse position.