EC 261/2004 places the responsibility for processing claims on the operating carrier. That is DL. It is not KL.
However, DL does use KL to process certain payments and that is up to DL. But, it does not affect who is responsible.
My experience with DL is exactly the opposite. Claim submitted for delay (EUR 600 and a bit of duty of care expenses). DL called within 2 business days to verify payment information and sent a check within a few days after.
I would suggest starting all over again and making the claim to DL. No mention of KL at all. APAC is a big place, so worth specifying your exact country, address, and preferred currency.