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KLM serves as Delta's representative for customer service in Europe, same as Delta does for KLM in the USA. So KLM will be your contact for this claim, contacting Delta will not really influence the processing time.
You might call KLM baggage office (you can call the reservation phone number to get the correct direct number) and ask for the claim number / case number so you can follow up later.
Processing is never quick, it can commonly take several weeks, before any offer is made. But Delta is usually pretty good at honoring their responsibilities.