In my experience, the airlines try to drag out the claim and wear you down. It seems to be a standard legal method.

If they don't respond after a few requests, then a letter before action and (in the UK) a small claims track action seems to be the only thing that makes them take notice.

Then they will likely pay up as soon as they see there is no further opportunity to ignore the issue. Only a few take the "Ryanair approach" and fight the case indefinitely.

There are firms that will persue the claim for you, and charge a percentage of monies recovered.

You may consider this a tempting alternative to reading up how to do it all correctly yourself.