This is not as simple as it is made to appear. The flight from which OP was denied boarding was KL-operated and thus it is KL which is responsible for any compensation, either under DOT's 14 CFR rules or EC 261/2004. DL is a complete red herring for purposes of mandatory comprensation (but not a customer service gesture). The fact that DL may serve as the ground handler for NA and may even cut compensation checks, does not change the fact that OP's complaint is to and about KL. A DOT complaint about DL will simply yield a response that the matter has been referred to KL as the operating carrier. Forget about alliances, JV's and immunity agreements. Those may govern how DL and KL interact, but they do not place any burdens on DL vis-a-vis this passenger in this matter.
This is where OP falls into no man's land. For "day of" IRROPS such as this, DL should have called KL and confirmed positive space. KL would presumably have authorized the transaction and awaited the e-tickets. It would not be uncommon for KL to have continued selling its flight (lots of other IRROPS on the East Coast) and by the time OP's e-tickets "arrives" he was a SB passenger.
DL's position is that it is not the operating carrier and thus not liable for compensation either under US or EU rules (EU in particular because for DL purposes it matters that this was a departure from the US). KL may say that it is not liable because it did not ever confirm positive space and thus OP was not IDB under US / EU rules. Thus, no reason to call for volunteers either.
OP is making a mistake by dealing with DL. If compensation is due, it is due from KL (even if it is DL which processes the claim and pays out the funds). If this is DL's fault because the JAX agents were too harried to get this done right, then that is a matter betweem KL and DL and not OP's worry.
Because this is a public forum and others face similar problems, there are some things to do which can lessen the risk and which, when one has time as here, will help:
1. Call the operating carrier of the new flight and confirm that it has you confirmed for positive space (and seats assigned).
2. While agents hate this, ask the agent handling the reroute to issue paper tickets. Agents hate this, will tell you that it can't be done. But, it can. This process forces the issue in the booking system and means that you have physical tickets (as opposed to BP's which are relatively meaningless).