How do you say thank you in Dane?

Dane is not a language. A dane is a person from Denmark.

In Denmark, people speak Danish, and in Danish tak means thanks.

You wouldn't say tak dig (thank you) - that's just not right in Denmark. If you want to say many thanks, you can say mange tak.

If people say tak to you, you can say selv tak (self thanks - a way of saying you're welcome).