The compensation here, if any, is for OP's expenses in reasonable purchases to deal with the missing luggage. Whether that had anything to do with the DL decision to have KL deal with the misconnect at AMS is not particularly significant.

The claim for missing luggage is made to KL at FCO and it is for KL to sort it out with DL. But, since DL handles CS for KL in the USA, if not resolved by the time of return, it will be DL which handles.

Ordinarily, one would ask KL for authorization to make a specified amount of reasonable purchases and those would be reimbursed upon submission of receipts. Even without authorization, it's likely that something would be paid.

Without any expenses and simply inconvenience, it's really hard to assess what DL (for KL) ought to do. Will likely be some # of miles.

Not to beat a dead horse, but cabin allowances are pretty generous. If careful packing is used, ought to be able to easily pack 5 days worth of stuff into carry-on, certainly the things you need on a cruise.