As a professional housekeeper/domestic cleaner myself, I'd clean the au pairs room, without thinking twice. It wouldn't cross my mind who occupied the room now/who used to occupy it before. It is a room in a dwelling that requires my professional services, of which I'm being paid for.
Culture clashes/pecking order insecurities are irrelevant. The client has a room that they want cleaning, so I clean it.
I'm quite friendly with some of my clients, but it remains at the forefront of my mind at all times that my services are a business arrangement, ones that I'm obliged to fulfil regardless of friendship.
Just to clear this up though, your cleaner is not your employee, she is a service provider. You don't class your regular window cleaner, oven cleaner, chimney sweep, gardener etc as your employee's, do you? (I appreciate you may not you the services of one or more of my examples, but you get where I'm coming from, right?)
I had an issue once where a client thought she could order me around like a lowly servant, speaking to me disrespectfully, and ecpecting to keep changing the goalposts of the service arrangement without prior notice and adding more duties into my already tight schedule, because she considered herself my employer and I should do as I was told.
I dropped her as a client.
There still seems to be a weird notion that hired cleaners should be whipped housenaids and be grateful for it.
I run my own business, and I do it bloody well. I take pride in my work. I quit a 15 yr career in somethong considered more 'professional' to do this. I'm educated and I'm not on the bread-line. I'm nobody's servant and I expect to be treated respectfully like any other service provider.
And as for "she chooses days to suit her, not me" Im afraid I have to do this too. Because, and I hate to brag, but I'm bloody good at my job! I'm sought-after with a growing waiting list of potential new clients, and my entire week is full to the brim of bookings. If I allowed every one of my clients to specify their days they would ALL pick Friday or Monday. I simply don't have the time to accommodate all my clients in those two days. I'd die of exhaustion and I'd never get home to my own family! So when I meet with a potential new client I let them know what day/time slots I have available. They are free to decline and look for another cleaner if I can't offer the day they want. But, because I'm a good cleaner, my clients have always been happy to take what days I can offer.
If you're not happy with your cleaners days OP, you are free to take your business to someone else. But just because she's a cleaner, it doesn't mean she should HAVE to give you the days that you'd prefer, like other pp's have been suggesting. She has commitments too... other clients, kids etc. You are not her entire world and your payments are not her entire income.
As in my previous comparison, you don't expect your other regular services providers, to give you the exact days you'd prefer. They will if they can of course. But if not you can both discuss other days or decline their services and look elsewhere.