Looks like no one actually bothered to search OPs dates. There's simply nothing lower than K available from the cities selected. It's got nothing to do with DL restricting inventory for the certs. I see L available for a couple of flights on MDW-TPA on 3/30-4/8. K fares are $521 rountrip for this route. A simple trick is to do a flexible dates flight search. If flights are pricing at $521 or higher you can assume they are K or higher fares and thus ineligible.
Thanks for the explanation. This is very helpful. Is that to say those eligible fares won't ever be available? In other words, should I give up?