This is a tricky one. I would have thought that this might be something more appropriately raised by the Social Worker, but if it's something which your mom needs to do, then my take on it, (Having been in the position of having to have this conversation with people more than once in the past) my recommendation would be:
- if at all possible, have a private, face-to-face conversation, rather than sending a note. It's much harder to ignore or dismiss a direct conversation, plus you can convey tone, and make it clear that you are sympathetic and looking to help, so you are less likely to be perceived as 'picking on' her.
- She will probably need to be fairly blunt. If the child is dirty and stinky, that's NOT just '2-3 days without a change of clothes', it's more deep searted than that. If the problem has developed gradually the sister may genuinely not notice, so the conversation may need to include something like "I could smell [child] as soon as he walked into my office / I came within 6 feet of him"
- be clear that your are sympathetic to her difficult situation and concerned for her and her brother, and frame it as a concern for the impact on the brother - it's reasonable to raise the fact that her brother is likely to be at risk of bullying if the issue isn't resolved, for example.
- if possible, speak to the social worker ahead of time about how social services may be able to help, so your mom is able to give advice/information about what help may be available or where she can go for help. "If you're having problems paying your water bill, SW may be able to help you apply for financial aid / provide support with budgeting / provide access to washing /cleaning facilities" (or whatever they can do)
- on a practical level, does the school have any resources which might help on a practical level? I recall in a similar situation (albeit with an older child) the school my mum worked for was able to give a child some spare shirts and things which came from unclaimed lost property, and he was told that he could use the showers in the gym, so it became possible for him to come to school a bit early, shower and change. If the problem is that sister has limited resources for doing laundry etc then extra clothes would help.
- are there any support groups for young carers which sister could be referred to? Or support / counselling regarding the issues which led to her caring for her brother? If the issue is new, it may be that something has changed in her life, and the hygiene issue is a symptom of that.