I
Me, Myself
pronoun, の adjective
I, my comrade (which is me), is I, it's myself, it's me. This is a weird way to say "I" and has a hint of arrogance to it.
わがはい
The reading comes from the readings you learned with the kanji readings.
吾輩が貴殿に久しく手紙を書いていなかったことをどうかお許しください。
Please forgive me for not writing to you in such a long time.