As noted above, per policy TK is not listed as one of the carriers DL will through-check to across separate tickets. You will find anecdotes of data across FT of some travelers who have had success having bags through-checked to non-exception carriers but you cannot bank on this. You can ask at check-in and the agent may, as DL does have an interline agreement to check bags through to TK if on the same ticket, but the agent is under no obligation to through-check to the TK flight. in your scenario due to not being on the same ticket. As such, allow sufficient time at JFK to claim the bag and recheck with TK (plus transfer terminals which you'll have to do anyway).