ESA Contributions Based is exactly that, it's based on National Insurance Contributions and unfortunately yir relative has not paid enough.
You mention "house money" so is that from the sale of a house? If they have too much in savings then again they will not qualify for income related benefits or Pension Credits.
Depending on her age, she may think about claiming Disability Living Allowance (if she needs help with mobility or personal care) or Attendance Allowance.
With regards to the Discretionary Housing Payment mentioned by dot, she will need to be in receipt of Housing Benefit or Council Tax Benefit and from what you say, she is not getting either.
You say Jobcentre Plus has been unsympathetic but there is little else they can do in these circumstances as they have to award benefits based on the relevant legislation.