My sister passed away four weeks ago, how to tell mum?

Mum is 86 and has dementia. Dad passed away 20 years ago, and she still sometimes waits for him to come home from work. My sister and I would remind her of his passing. Our brother died about six years and sometimes we would have to remind her of his tragic death. Now my sister died from a brain tumour. We had a lovely funeral send off for her, but excluded mum because I thought I could not bear to remind her of another family member's death. Yesterday, Mum had been trying to call my sister to wish her a happy birthday which was on the March 1st, but her number is disconnected. Mum wants me to give her the new number. Should I tell her the truth? and then the consequent sadness and tears for me, and then forever reminding her of the passing of her eldest child. The anguish of my sister's death is enormous as we were very close. I am so tired and sad of reminding my mum about my dad and brother. I feel terribly guilty about not telling mum, as she has a right to know. The last few day when my sister was in hospital dying, I cried so much the thought of not having her around and how to tell mum, my gorgeous sister said "don't tell her" and at the time I thought it would be best not to also. And now if I do, how do I explain the funeral, the burial, and how she died (she was cremated and ashes scattered to sea). Mum still lives at home in the country and I work in the city but I visit her every Saturday to do shopping, cleaning, have a meal together. I having been dreading the day when mum would realise she has not heard from sis (sis would call every day in the morning and I would do the evening calls) and that day has arrived. I am making myself sick thinking of what to tell her... pretend to be my sister on the phone (we sound the same); tell her she went to the funeral but has forgotten, or what? The last time mum saw sis was at Christmas and was shocked at her diminutive (she does not remember) and sis told her she had cancer and had 12-18 months to live but the disease took her early, well in my mind too early - it was almost 12 months since diagnosed. I miss her terribly. I don't know how I will cope without be best friend, adviser, critic, confidant and everything else a sister can be. I will be calling mum tonight to see how her day was and hope she does not ask about sissy.