When you are moving and register your change of address to Copenhagen, you will be asked to choose a new doctor as well, free of charge. When you use this online service you will find a list of doctors in your neighbourhood (within a radius of 5 km from your new home) that you can choose from.
The new health card with your new address and doctor's information is free of charge, and will be sent to you by mail within 4 weeks.
If you do not have a NemID, you can:
Transfer the money to account 0216 4069177506 (Danske Bank). You can pay using online banking. In the message field you must remember to enter your CPR number, the identification code or name and address of the doctor plus the word 'lægeskift' or ‘laegeskift’.