To add some additional context, the character appears to have family problems (particularly father problems). Her birthday is on Christmas, however her father never celebrates it with her.
I don't quite understand what 実感 is referring to in the below.
「……お父さん」
見知らぬ人だ。
わかってる。
ちょっと口にしてみただけ。
手にはケーキの紙箱と、赤と緑のクリスマスカラーのリボンでラッピングされた袋。
ケーキにプレゼント、お約束中のお約束で、子供のためという父親の意気込みが見て取れた。
「ひねくれたこと考えてるな、私」
そういう言葉の端から実感が欠けていく
実感 is "sense of reality", and in this context, it refers to the sense of realization that he is her father. 欠ける is "to lack/miss/disappear", and ていく is "over time" and/or "away from me". So the sentence is saying that, because she is seeing the situation sarcastically, she is becoming more and more unsure if she can accept him as her father. The more she is seeing his blatant effort, the more she is seeing him as a remote being.
