>Reflects well on the EU that she was elected.
It would, if hers was an elected position :P
The European Commission is a bit of a strange beast. As is the rest of the EU, to be honest.
The Commission consists of the President of the European Commission, the High Representative of the Union for Foreign Affairs and Security Policy (effectively its foreign minister), and 26 other commissioners.
The President of the European Commission is elected by the European Parliament[0], and the High Representative is then chosen by the European Council[1], with the consent of the Commission President, who cannot be from the same country as the President and will be one of the Commission's Vice Presidents. The other 26 Commissioners are then appointed by the unrepresented Member States, so they each have one Commissioner.
The Commission President then assigns each of the Commissioners (except the High Representative) a portfolio and then the entire body is subject to a confirmation vote by the European Parliament before it can take office.
[0] It can't just vote for anybody, though, the European Council technically has to propose a candidate "taking in to account" the results of the results of the election to the European Parliament, although in 2014 (the first time they had the opportunity since the rule was introduced) Parliament successfully managed to establish the precedent that the Council must nominate the designated candidate ("Spitzenkandidat") of the European Political Party[2] that won the most seats at the election in the first instance.
[1] The European Council is the body consisting of the heads of state and government from the EU Member States, not to be confused with the Council of the European Union, which consists of ministers from Member State governments, and the Council of Europe which is a distinct organisation.
[2] European Political Parties aren't political parties in the same way they are at national levels - they are more coalitions or federations of national political parties - and they are also distinct from political groups in the European Parliament.