What should I do about the damage to the electricity meter reading logbook that I share with my neighbor, and should I tell her about it given my fear of her thinking ill of me?

It is not permissible for the sister who is asking to assume an amount and demand it from her neighbor as her share of the electricity bill, because she does not know the extent of her neighbor's consumption, and thus she would be taking from her what she is not obligated to pay.

The correct action is to inform her that the notebook is lost, and then they should agree on paying the bill. The fear of being thought ill of is an imagined harm, and it is lesser than the harm of taking someone else's money without a rightful cause.