If she got paid by your dad (via you as Attorney) to look after him then she wouldn't be able to claim carer's allowance as well, would she? You either get a state benefit as an unpaid family carer, or you are a self-employed carer for a member of your family. Both ways are legal but not at the same time IMO.
Your sister's list of 'expenses' is a red herring. Expenses as such should be those relating to your dad, e.g. if she paid for something for him or his household that he/you would normally pay for, then she presents a receipt to you and you reimburse her for it. What she means is that she has her own personal living expenses that she would normally expect to pay for out of her earnings.
Perhaps a compromise would be for your sister to have a part-time job outside the home and to be paid an hourly rate for the time she looks after your dad when the live-in carer is not on duty. I must say, from my experience of employing live-in carers, it can be a blessing to have a family member on the team but can also create problems with knowing who is in charge and responsible for which duties.
Does your carer work to a written care plan? Your sister's input needs to be clear on this plan. It appears ATM that things have developed ad hoc and you need to clarify and formalise the situation. £100 at week is not unreasonable, provided that your sister's input is substantially worth that sum. If you employed a second part-time agency carer I would think you would have to budget for at least that amount. It really comes down to the question of whether your sister puts in sufficient time and effort to be worth the money.
I think you need a Plan B in reserve. If your dad needs support from more than one carer, how will you sustain that level of care at home using his current income? If your sister decides she cannot afford to continue as at present, or decides she doesn't want to do it long-term, then you would need to budget for alternatives.