I'll try and claim via insurance. The most frustrating thing was the lack of comms.
I hope you're finding some success. Dealing with more then one carrier would no doubt complicate matters, when lack of communication just compounds the frustration.
From recent experience, Qantas, their JFK ground staff, their marketed travel insurance, compensation and communications were a total mixed bag.
Apparently an airbridge operator damaged an engine, which was not identified until all passengers were boarded. After 3 hours, passengers were informed the flight could not operate and would have to disembark.
That obviously caused ground staff to manage passengers quickly, to find accommodation, transport, food for everyone from a full 787.
They appeared inexperienced but managed the situation calmly, always stating the Qantas options. Accept a found hotel room (which Qantas would pay for) and make your own way to it (Qantas would not pay for travel costs) and a $30 food voucher. Or find your own accommodation (Qantas would only reimburse $300AU), make your own travel arrangements and accept a $30 food voucher. Most passengers opted to accept the hotel found by Qantas, only to later discover the hotels were as far as 25 miles away and the taxi fare cost to get there, over $100US. To find any hotel in New York (or London, Hong Kong, Honolulu) for only $300 is these days impossible. The Qantas compensation is totally unacceptable. When delayed a further day, I opted to find my own accommodation at the airport, which was very expensive but far more convenient. Qantas had no argument with reimbursing me the $300 and did so quickly after I lodged the claim.
Qantas rebooked passengers and informed them via text messages but from conversations with other passengers at the airport the next day, these messages about rebooked flights came in after the rebooked flights had actually left JFK.
Again, ground staff were helpful (but appeared out of their depth), it was witnessed that they were calling the call centre in Hobart to find flights to rebook individual passengers and the whole process was very slow.
To be fair, a grounded plane and having to rebook all passengers, would not be easy. I felt that Qantas did its best to help but the ground staff at JFK and the messaging comms just reinforced the confusion.
When making a claim from Qantas Travel Insurance (underwritten by NIB and an even higher organisation) from a worldwide comprehensive policy, for reimbursement of additional expenses, I was informed in very legalistic terms that a claim would not be accepted. A subclause in the Qantas Travel Insurance policy stated that they would not reimburse costs 'due to mechanical faults in transportation'. Qantas had issued a letter to passengers at the outset that the flight had been cancelled due to 'a complex mechanical fault'. (I would like to argue it was structural damage caused by an airport worker but it seems Qantas was covering the incident from the start with language which would prevent insurance compensation).
Be very aware with travel insurances. Read all the fine print. Not many do have that subclause exemption but if they do, why would you ever buy the insurance? Most people take out the insurance to cover unforeseen medical costs or unforeseen added expenses caused by travel delays, cancellations etc.
A final point, I then tried to make a claim through my credit card provider and with hardly any questions asked, they reimbursed my added costs within days. Lessons to be learnt about dealing with Qantas, their travel insurance, their compensation limits and the anxiety of such an event.
Apologies for the long post, good luck jjjaaa.