Unfortunately, the adapters from the Wahoo Kickr Core do not support mounting this system. I had a look and, according to my very limited knowledge, I need an adapter similar to this one:
@tmcampelo The metal adapters should say QR. It looks like you have the correct one mounted based on the picture. You may be able to take the skewer from your wheel and reuse it on the Kickr or just purchase a generic skewer at any local bike shop.
Contact support for additional details. There is also an installation video online that should help.