申し訳なさと不甲斐ない自分に is not attached to 重ねて. 申し訳なさ and 不甲斐ない自分 are both the causes/reasons of 謝る, thus, に has a function similar to ~について in that sentence. 重ねて is an adverb, meaning "again/repeatedly".
Thank you very much for your reply. Is it a common usage of partice に? It is a first time I've seen a usage like that, and no one of my grammar books appears to have this usage written in it...