The issue has nothing to do with who actually lost the bags or with alliances. It is an IATA convention that the last delivering carrier is responsible to the passenger. If the bags are recovered, it will be KL which delivers it to OP. If the bags are lost/damaged, it will be KL who reimburses the passenger. It will then be for KL and DL to slug out responsibility, but that has nothing to do with OP.

Generally speaking, KL would pay for necessities until the bags arrive. But, here, unless KL can tell you that they are headed your way, I would call KL and ask for a commitment allowing you to replace presents (recognizing that what you want may not be what is available). The issue is time-sensitive and you might just might get a KL agent to go along.

As others note, almost all bags turn up fairly quickly.