The fact that other carriers' aircraft were operating has little relevance to KL's operations, especially at its worldwide hub. If KL won't pay, your option is to sue. You live in the UK, so you can use the MCOL process. All the forms, including the letter before action, are available online. Rather than mucking around with back-and-forth, send the proper letter before action and then, if KL does not pay, you are free to file with MCOL.
The burden will be on KL to prove an extraordinary circumstance. But, don't be surprised if it can do so.