I have no idea. I mean, I'd like to know the answer to that question.
Well, beats me. I'd love to know why myself.
ご質問ありがとうございます。
① "I have no idea."=「全く知らない。」
"I mean, I'd like to know the answer to that question."=「というか、私がその質問の答えを知りたいよ。」
② "Well, beats me."=「さあ、さっぱり分からない。」
"I'd love to know why myself."=「なぜなのか、私自身が是非とも知りたい。」
ご参考に。