You do not need to have any vaccinations to visit, and being part of the European Union means that travellers from the UK do not technically need specific medical insurance for travel to Denmark, as there is an agreement in place that guarantees reciprocal medical care in an emergency although you do need to have a valid E111 with you. This should be done prior to travel (forms are available from post offices). However, you should still have comprehensive travel insurance when you go abroad, in order to cover repatriation expenses in the event of sickness, and theft.
The hospitals in the city are of an excellent standard, and offer a full service for travellers.