In the hadith of the man who prayed badly, the prophet taught the mandatory aspects of the salaat, without which your prayer is considered invalid ("Go back and pray, for you have not prayed."). The recitation itself is explicitly commanded; how loud this recitation needs to be and how many suwara need to be recited is not mentioned at all.
Reciting a second suurah after Surat al-Fatihah for the first two raka'at of the prayer is well-established as the sunnah of the prophet. However, this recitation is not one of the obligatory parts of prayer:
...he who recites Umm al-Qur'an (i.e., Surat al-Fatihah), it is enough for him... (Sahih Muslim 4:780)
Reciting the Qur'an loudly during some salawaat and silently in others is also sunnah, especially when leading a congregational prayer, but it too is not obligatory:
...Sometimes [the prophet] would recite loudly and sometimes softly... (Sunan Abu-Dawud 1:226)
Whether you decide to pray the missed raka'at as if they were at the start of your salaat (by reciting out loud with an extra suurah) or at the end of your salaat (by reciting silently with no extra suurah), it is important to note that your prayer will be valid either way. This being the case, it is better for you to pray in whichever manner you are most comfortable, so as not to distract yourself with unease during the remainder of the salaat.