Could you offer in a friendly way to give a hand maybe once a week. Sometimes a helping hand offered in a way that does not come across as interfering may well be appreciated. Some people by their nature are more messy and mucky than others. I would be quite worried by the slime in the cup, and the black mould, these are both health risks. One way to keep these cups clean and safe quite easily is to buy some good old fashioned Milton solution, used for sterilising babies bottles. It is available in liquid form or tablet. You don't need to buy a special sterilising unit, just a suitably big enough plastic container with a lid and its straightforward and easy to use. Obviously this has to be done regularly and kept out of the baby"s reach.
A solution of white vinegar and water in a spray will help with the black mold. It actually stops it multiplying if used regularly. Condensation is often the reason for it, and not opening windows enough. I would be concerned in your shoes, but your daughter may also be depressed as well, and this could be contributing to her lack of motivation. Hopefully she will accept a little help from you and get back on track.
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what’s going on with the dishwasher?