My friend helped you and me^^;

He said…

「私とは、なんじゃいな」

　日本語はご存知の通り「主語」が文の要素（必須）ではないため文脈によって省略（厳密には「必要ではない」ものだから省略ではありませんが）されます。

　この「私とは」の部分は「主題」ですが、主語を補うと「『鏡の中の見かけの違う人物が』私であるとは、いったいどういうことなのだ」となります。この「鏡の中の見かけの違う人物が」という主語が文脈から補われないと理解できないでしょう。

　「なんじゃいな」とは「何だろう」、「何であるのかな」の口語的音便形です。
　意味的には「なぜそうなる？」や「どういう意味だ？」などを広く指します。

The person who looks quite different from me is me?（ここまでが「私とは」に相当）
What in the world is it?
=How come?
=How can it be true?

In the Japanese language, the "Subject" is not an essential element for a sentece and is very often left unsaid as far as it is understood from the context.
In this case, the subject is "the person who looks quite different from me".
"Watashi de aru to wa" can be translated into "(the fact) that the person who looks quite different from me is me". The particle WA is a topic indicator, not a subject indicator, and WA can be directly interpreted as "when talking about..." or simply "about..."

About the fact that the person who looks quite different from me is me.
This is the literal translation from "watashi de aru to wa".
"Nan jai na" is a colloquial form of "nani de aru no ka na", which may widely mean "what is it?", "what can it be?", "How come?", "What does it mean?", etc.

So, the original sentence can be understood as "The person who looks quite different from me is me? How can it be true?", or something like that.