Two of the passages that must be recited in Lashon Hakodesh are Birkas Kohanim - the blessing of the Kohanim which is recorded in this week’s Parsha, and the Parsha of the king, referring to the public reading of the king at Hakhel.
The Sages discuss the source from where we derive that these sections must be said in Lashon Hakodesh. Birkas Kohanim is recited in Lashon Kodesh because the Pasuk says “thus shall you bless” - כה תברכו, alluding to the fact that it must be recited as recorded in the Torah.
The Sages do not teach the source for why the Hakhel passage to be recited in Lashon Hakodesh.
A possible explanation; the other passages in the Mishna were recitations performed at different times. Whilst they quoted words from the Torah, they were not read from the Torah. Therefore a source is required for why they must be recited in Lashon Hakodesh. An example of this is Birkas Kohanim.
In contrast, the Mitzvah of Hakhel (as the Torah itself specifies) is to read from the passage from the Torah scroll itself, which would obviously be in Lashon Kodesh.