Why would she do that unless it was true?
You don’t know any better than anyone else that it wasn’t the cleaner who took it. If it gets removed from her record then if someone else reports something missing they’ve not got a previous incident to refer to in order to draw inferences she’s done anything wrong.
God forbid that some of you should end up on a jury any time.
As things stand the OP has no evidence on which to base her accusation. Yet in the heel of the hunt someone stands accused, and her file reflects that accusation.
This is not just. The OP invokes the principle of justice, a legal concept, while conveniently forgetting that the law relies on evidence that is sturdy enough to dispel reasonable doubt, not circumstantial evidence, which is all the OP has to go on along with a good many assumptions about her own family. If we are to get all legal here, with an invocation of 'justice', the prosecution has to prove guilt. Can the OP do this, in the interests of 'justice'? No, she can't. So she needs to back down, eat humble pie, admit to herself that she was hasty and wrong, and do something just instead. You can't let an accusation hang over someone when you don't have solid evidence. The accusation must be withdrawn.
Merely because someone else experiences a coincidental loss while the cleaner is employed doesn't necessarily mean she is the person responsible either. But she could end up with two strikes if another client is hot-headed enough, and sufficiently invested in the idea that "there is corruption everywhere" except in her own family, to call an agency and state what might be outright lies. You don't have the right to be wrong if that means you are putting someone else's livelihood in jeopardy.
The OP is not responsible for what happens down the line if someone else experiences a loss. What you are calling to mind here is basically a posse of vigilantes all holding the line against marauding cleaning women.