Deeply sighing ἀναστενάξας involves one's breath. The word πνεῦμα has the meanings breath and wind, as well as spirit. Translating τῷ πνεύματι αὐτοῦ as "in his spirit" implies that Jesus gave no visible appearance of sighing. This doesn't seem to match ἀναστενάξας, which means "deeply sighing."
In English "with his breath" is somewhat redundant with "deeply sighing." Thus, NIV1984 translates Mark 8:12:
He sighed deeply and said, “Why does this generation ask for a miraculous sign? I tell you the truth, no sign will be given to it.”
Many translations translate τῷ πνεύματι αὐτοῦ "in his spirit."
Note ר֥וּחַ in Hebrew means the same thing as πνεῦμα in Greek. Here it means breath.
בִּדְבַ֣ר יְ֭הוָה שָׁמַ֣יִם נַעֲשׂ֑וּ וּבְר֥וּחַ פִּ֝֗יו כָּל־צְבָאָֽם׃ (Psalm 33:6, BHS)
By the word of the LORD the heavens were made, and by the breath of his mouth all their host. (Psalm 33:6, ESV)