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Where did you get the new KLR -- are you sure it works? I've picked up several dead DME and KLR boxes from ebay over the years that were claimed to work. If you're sure it works, I'd check the harness wires between the KLR and DME for breaks or shorts.
Have you checked the harness to confirm clean connections carrying the ignition pulse to and from the KLR? How/where are you jumping the KLR when it runs?