My braces are killing me.
矯正が痛くて死にそう。
killing me = (痛くて)死にそう
My wires are poking my cheeks and it's killing me.
ワイヤーが頬の内側に刺さって痛くて死にそう。
poking wires = 矯正で口に刺さる飛び出たワイヤー
The wire on my braces is poking into my cheeks and it's really painful.
The wire on my braces keeps stabbing my cheeks and it's very painful
「矯正が痛い」は単純に「my braces are really painful」や「my braces are hurting me」になります。しかし、痛い理由が説明したい場合には、こちらの英語を使った方が分かりやすい:
The wire from my braces is poking into my cheeks and it's really painful.
The wire on my braces keeps stabbing my cheeks and it's very painful.
The wire on my braces hurts the inside of my mouth.
「The wire from/on my braces」=「歯列矯正のワイヤー」
「is poking into my cheeks」=「口(ほっぺた)に刺さって」
「and it's really/very painful.」=「痛い」