Charge-back. Seriously. Your full contract has not been fulfilled and so long as you keep a diary of contact with the company and can reasonably demonstrate that you are being stonewalled with incompetent staff or unwilling company then you will be fine. Note that charge-backs will backfire on you very often if your claim is frivolous or not well supported by your own diary of the chain of events and proof (email, faxes, letters or whatever). Note dates, times and CSR's company number or name when speaking verbally on the phone, makes an enormous difference and I do this out of habit now with all call centre comms to all businesses (not just airlines).
Once the charge-back succeeds the company can contact you, if they are not too incompetent, to work out reasonable payment terms for the services that were actually provided.
None of this has to be nasty or angry, in fact, it will all work better for you if its completed in a businesslike and fairly dispassionate way.
As the quality of service from the big call centre's drops further and further, due to lack of staff, training, policy or whatever its in the businesses interest, short term financially at least, to have you give up and go away. Ultimately its a bad move for them of course because an upset customer probably won't return, but they don't see this in the immediate term.
Rescinding payment against an unfulfilled contract often gets their full attention, which is really all you want.
