Eve
I would check what your sister-in-law has been told as 59 is not an age that I would associate with ESA ending. Her ESA should continue in payment, as long as she has an award until she reached State Retirement Age.
It is likely that she will be migrated to UC if her ESA is Income Related before she retires.
Her ESA will be replaced with her State Pension, if her income is low enough then she may also qualify for Pension Credit.
Gordon
Nothing on this board constitutes legal advice - always consult a professional about specific problems