knakered pay in lieu of notice means you dismiss someone with effect immediately, and instead of requiring them to work their notice, you end the employment then and there and pay them a sum to compensate for the loss they have made as a result of not being given the opportunity to work their notice.
It's easier to do if you have that clause in your contract but you can still do it without. The reason you would need to do this rather than just give her her month's notice is because if you just terminate her employment with a month's notice now, her notice period would take her over a year's employment, and she would be able to claim unfair dismissal.
Your options are:
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Dismiss her immediately, make it very clear the last day of her employment is today, and pay her one month's salary in lieu of notice. Put this in writing, again making it very clear her last day of employment is today and that you are paying her a month's salary in lieu of notice.
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Make her redundant, it's not clear to me that this is a genuine redundancy, so that's a bit risky.
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Discipline her, give her a written warning. You wouldn't be able to dismiss her for gross misconduct because of the reasons I mention above, but you could put her on a final written warning and state that any further unacceptable behaviour in the next 6 months is likely to result in dismissal. If she then steps one foot out of line during that time, you should be able to dismiss her without problems.
The reason it's so complicated is because to be honest you've left it very late, you've not acted when these things have happened and you've left it very late to be able to just dismiss her with no legal risks, because she's got almost a year's service.
If you choose option 1, it's very important you do it as I've said, making it clear her employment has ended immediately, and that you are paying in lieu of notice, you should meet with her, state those things and give her a letter saying that as well. This option isn't without legal risk as it's so late, but it gets rid of the problem straight away. I don't know whether she is the type to seek legal advice, she may do and may challenge it. I think you'd be ok but it may be a bit of hassle.
If you choose option 2, you'll have to go through a process, meeting with her, stating that you no longer require a full time nanny, you are reducing your requirements to 3 days a week, offer her that formally as an option or redundancy, give her a chance to go away and think about it, and respond to you, if she says no, you need to pay her one month's notice and put in a letter that she is redundant. You could keep her for the duration of her notice period with this option if you want to.
If you choose option 3, that's probably your safest legal option but would involve keeping her at least for a while, although she would have to improve immediately.
I apologise for this being complicated, as I've said, unfortunately the timings and delays have made it very difficult.