If you find the cultural differences tough, choose your next AP from your own nationality.
The way you describe her "this idiot", she is a young girl in a different country, I thought the point was an AP came into your home and was treated almost in the role as a family member...
Different countries have different ways of doing things, or may not have experienced certain things at all.
The shower thing is shit, I would be pissed off also but would also keep into perspective we do all make mistakes.
Do you know what I thought when I read that she said it wasn't her? That doesn't surprise me, it would be hard enough to admit something anyway, but I just have a gut feeling that admitting it to you would be harder!
Or maybe she thought she turned it off, that's why she denied it.
Plus what do you mean saying the minimum of sorry? She said it once, done what do you want her on her hands and knees begging for forgiveness for the next 3 months? Also different cultures respond to things like apologising in different ways.
"That this thing"..., do you know Im a gob shite, can be a bit of a hot head, Im certainly not a drama lama and can understand the venting (I do often). However your choice of words and the way you describe her actually makes me feel very uneasy.
There is no way she is not picking up on the way you feel (even if you intend to come across as different in RL).
As I have said before this is a person you have brought in your home, if one of your children were doing this within anthers home in the same circumstances, how would you feel if you knew they were being referred to as a "thing"?
The lazy lazy thing to your DC, not nice. Again different cultures act differently, maybe she thought it was a joke? You addressed it, its sorted.
Did you hear it happening and address it right away?
Just your OP is a bit unclear and almost reads like you heard it happening, but only tackled after your DD broke down about it.
You talk about what an AP should be doing, surely that with the hours of work etc are all in the contract. If you don't think her way of doing something is working, talk through it with her ffs.
You reasonably asked her to go to her room, because you were so upset she wanted to watch the TV, and the sight of her makes you furious.
Get a grip, she has an accident breaks her TV, goes to watch one in another area of her home (as that is what an AP arrangement should be like), you want her and tell her to go, nice!
Oh "that thing" has come back again, I actually hope you or your DW take some emergency time off, sort out childcare long term, and let this young lady go.
Your so concerned about her behaviour you need to write her a list to turn off cooker, but are still happy to have her in your home, putting all at risk as it suits you due to your work arrangements.
You are either so concerned she goes and you deal with it, or your not, work should not be paramount to your families safety!
"I've also told that she's on a short order and if she messes up one more time, she's out."
You do know she will after saying this don't you, imagine the anxiety after that being said to you.