"...for a holiday with me in Denmark" Would sound better in English:
"...to spend a holiday with me, in Denmark"
I'd use "expenses" instead of "economic charges" and I'd also change the syntax. Instead of:
"During the visitor´s stay with me in Denmark I shall assume all economic charges that may occur."
I'd say:
"I shall assume all the expenses that may occur while the visitor is staying with me, in Denmark."
"Jeg vil tage mig af de nødvendige økonomiske utgifter" - nødvendige=necessary, hence: "I shall assume all the necessary expenses that may occur etc.". Not a big difference, but possibly an important distinction.