They have been with you for a short period of time, so are they still in probation? At some point you need to decide if things are working out or not. Failure to turn up at work is clearly a problem, regardless of the reason why they fail to turn up. It can't go on for ever, so if their health is so bad that they can't do the job, then they may need to work elsewhere. For now, manage the attendance but as part of probation review you will need to consider the amount of time they have had off, is it acceptable, is it likely to continue.
ACAS: Managing Attendance (PDF)
Can the job be adapted in any way to enable them to work whilst ill? A GP may well issue a Fit Note which states "may be fit for work" but you need to decide what parts of the job could be done whilst they are ill.
Tiredness may be an issue but does that mean they can't work? I expect that many of us nannies have had bad days when we were not feeling great and we still went to work and spent more time at the house rather than going out and about. With sinusitis there can be pain involved, but isn't that what pain killers are for?
GPs don't issue a Sick Note any more. It has been replaced by a Fit Note. The Fit Note will say if the GP feels that they are not fit to work - thus SSP can then kick in.
SSP kicks in once they have been sick for at least four days in a row (including weekends, bank holidays and days that they do not normally work).
HMRC: SSP Calculator
Employer Handbook about SSP
If you use a payroll company, talk to them about how SSP is calculated. Make sure your nanny also knows that SSP will kick in and that it isn't much money.
Does your contract mention SSP? Does your contract make provision for your employee to have any Sick Days as paid?