If is impossible to protect against your own family: the have hardware access to your computer, they can intercept all your paper mail, they know all details about your life, etc. So they are the perfect identity thieves.
Some Danes think they are clever and scan the paper card and store it as an image on their computer. Some people are just impossible to make a secure access system for.
The upside of the downside is that if anybody gets hold of your login details, then there is a single place to stop them instead of having to change 20 logins.
BTW. "NemID" translated to English is "EasyID". Within the next year, a hardware dongle will be available (e.g. for users who often login and uses up all the codes on a paper card in no time).