Did you report the full facts at the time to BA? Did they see the damaged case etc? BA, like all airlines, carry insurance to cover such situations and you should have been asked to fill out the claim form by the Customer Service person that you reported the damage and loss to, at the airport. You say you reported it "immediately" so, hopefully, you did report it to BA staff at the airport?
Airlines are always reluctant to pay out as the amount is quite likely to fall within their retained deductible. i.e. the airline will have to pay itself rather than their insurers.
Somewhere in your ticket, in the small print, it will proboably tell you what the carriers, (BA in this case), liability is and I doubt it will come anywhere near $2470:00! But they should definitely be paying you something, go after them.