The main issue, apart from proof, is whether there was something resembling an agreement between the parties. Since you mention a (previous) contract, I assume that means that she was bound to pay for service for a certain period of time or pay a cancellation fee for early termination. That strikes me as sufficiently unusual that it might actually be illegal (power being a highly regulated industry), depending on state. Assuming there is some such fee, or at least the hassle of cancelling the service, then she would be getting something of value. The other side would also get something of value to them (avoidance of hassle, service interruption, possible credit check issues). So you have "consideration", an essential element of a contract (which the other side breached).
What she presumably has to do is "not cancel the service", and what they have to do is "pay the bill". She apparently did what she was supposed to do, and they didn't: a breach of contract. She was harmed substantially (there may be some problem quantifying the damage to credit rating, even if they eventually paid the bill). Were it not for the matter of compensating her for damage to credit score, this should be a simple enough Judge Judy type matter where she sues the other party in small claims court. Let us assume that the understanding was informal, i.e. there is no written contract. That doesn't matter, what matters is that each side understands what they are getting and what they have to do. It is simply not credible for the other side to say "Well, we thought that she was just paying out electric bill because of her deep and abiding affection for us". So even if they deny there having been such an agreement, they acted as though there were such an agreement (they did not get the account changed so that they would be responsible). It would probably be mandatory to consult an attorney to decide how and whether to sue.
There is some possibility that the arrangement would not be considered to be a binding agreement, that is, the parties had no intention to be legally bound, within a family. That doctrine is more prominent in England than in the US, and there certainly isn't any doctrine here that there are no binding contracts between family members (I can find no case like Balfour v. Balfour in the US, which held that it is rebuttably presumed that domestic agreements are not legally binding). Even in England (Bestwick v. Bestwick) family members can be held to the terms of a contract. Whether or not such an argument would gain traction in court would depend on objective facts, reducing to the question "do the facts indicate that this was a gift?". Such an argument might be more credible for an ostensive agreement that was made in the course of an amicable family relationship, but is not very credible in the event of a divorce or other break-up.