new job problem

Document everything.

In these days of litigastion for the drop of a hat it is always wise to be always on your guard. I hate to say it but that is the way of the world these days.

You need to have processes in place to protect yourself from this sort of allegation and also to protect your staff.

They need to understand those processes too. As someone said above- clear objectives, weekly 1-2-1s . If someone is not performing you need to be clear and explain that to them, put in place improvement goals and make sure they understand that if you have to go down the route of dismissal you will have covered all your bases.

I have been there and I know how stressful this is . Maybe take time to consider if management is really your forte too? Not everyone makes a good manager.

Yes Lyndylou is right, ensure everything is documented, be clear about the objectives.
A management role will always bring these issues, as others have said you are not a friend or co-worker you are their manager. If it's having a detrimental effect on your life perhaps you need to reconsider your role, is it worth the stress ?

I think you need to get all the cleaners together, explain/show them exactly how you (and the management, who should back you) expect things to be done. That way no one will be able to say you are picking on them and everybody will know where they stand.
You are their boss first, friend second.
Always found it best to set things out clearly with staff.

Weekly 1-1 chats, clear objectives, lay out what you expect of her and document if she is achieving it or not. Doesn't have to take long, probably best if you can keep it simple and succinct, but make sure everything is written down and, most important, that she signs it as a clear and true representation of the meeting before she leaves the room. I would write up bullet points before hand.

I used to do this sort of thing a lot, now I'm temping I see major problems arising because managers are being too nice and not dealing with issues early enough,

I'm not sure if there are cleaning courses available but I will ask, thank you. There is a disciplinary system

This might seem a little brutal but my advice would be "stick to your guns" - they (your employers) thought you were "up to the job" so prove them right. You're the new guy on the block and he/she resents it - probably thought they should have got the job themselves - so it's just a question of staying true to yourself and believing in the new you - I bet your new husband believes in you so why not prove him right? Confidence is the key - whether you have it or not just pretend that you have - believe in yourself AND follow HildajenniJ's advice.

Sounds like your staff member needs to go on a training course. Does the home arrange these for the domestic staff? Could you ask the manager for a cleaning course for all the cleaners so that she doesn't feel singled out? Maybe something with a certificate at the end. Do they have a disciplinary sytstem there?