According to the Hanifi school, a pilgrim in a state of ihram who, for a genuine reason, violates any of the restrictions of ihram, like shaving the head or [for men] wearing sewn clothes because of cold or heat or something else—with the exception of having sexual intercourse—may do so, but he or she will have to slaughter a sheep, or feed six needy people (each to receive at least the equivalent of half a saa` in measure), or fast three days to atone for it.
Allah Almighty says: “Perform the pilgrimage and the visit (to Makkah) for Allah. And if ye are prevented, then send such gifts as can be obtained with ease, and shave not your heads until the gifts have reached their destination. And whoever among you is sick or hath an ailment of the head must pay a ransom of fasting or alms giving or offering. And if ye are in safety, then whosoever contenteth himself with the Visit for the Pilgrimage (shall give) such gifts as can be had with ease. And whosoever cannot find (such gifts), then a fast of three days while on the pilgrimage, and of seven when ye have returned; that is, ten in all. That is for him whose folk are not present at the Inviolable Place of Worship. Observe your duty to Allah, and know that Allah is severe in punishment.” (Al-Baqarah: 196)
If it is necessary for you to wear braces in general, there is nothing wrong in wearing them during Hajj, and you are not required to atone for it in any way.