There are likely 2 explanations for the OPs bill:
1. Someone living in or with access to the house is making the calls when she is not around and will not admit this. OP seems very sure this is not the case.
2. It is possible that someone in a neighbouring house has the same model phone working on the same wireless frequency (this could be by accident, but possibly by design - I'm sure its not difficult for someone to pick what frequency they want the phone to use - most cordless have this option as it is possible that by coincidence a neighbour is on the same freq so the user may have to change theirs). Their wireless handset is making calls thru the OPs phone connection rather than their own.
Either way, I would expect the phone company to refuse to refund. They will obviously claim that no. 1 is not their fault or concern. They will also claim that in the case of No. 2 it is the OPs responsibility to ensure that the phone equipment they are using is secure and not operating on the same frequency as similar equipment in the area. You may argue that the phone was also supplied by Eircom, but its likely that the terms & conditions of the purchase absolve them from blame if the phone is working correctly - they will say that its up to the user to ensure the frequency is correct.
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Note that the particular model of phone has a 300m range - this could potentially cover 100's of neighbouring phones !!!!!