For readers more familiar with the US, the closest analogy is European Parliment == Congress, European Comission == Executive Branch [2]. Far from exact, but will give you a flavour of the reason the parent comment is so off base.
[1] In the countries I'm familar with they're all directly elected. Not sure if that's the case in all member states.
[2] Structure of the EU: http://publications.europa.eu/code/en/en-390500.htm
And while I'm on the irrelevancy track (apologies) - how about an unelected foreign minister with, as reported by observers, no demonstrable talents and certainly no mandate. What are we coming to?
I'll note that the public in Denmark did not vote for her to be Deputy Prime minister, they voted for her to be an MP, being a Deputy Prime minister is a nominated position. There are lots of nominated positions in representative democracies. In this case, the position is nominated by the elected government of a host country, then the selected nominations get voted on by the elected EU parliament. It is a bit like the process of nomination to the US supreme court, only the whole set gets voted on.
There is an argument for the Commissioners to be initially selected by election in member states rather than through nomination and that was raised in 2010 by European Parliament President Jerzy Buzek, though I don't know how it is progressing.
She couldn't be a foreign minister though, if you meant it in the other sense, as she is definitely from Europe and is in an EU post.