What historical event (if any) actually took place on the 25th of Kislev to cause that date to be the commencement of Chanukah?
This might be basic but it is quoted in almost all halachic texts.
Rambam Hilchos Channuka 3:2 ,Gemara shabbas as well. וכשגברו ישראל על אויביהם ואבדום בכ"ה בחדש כסליו היה ונכנסו להיכל ולא מצאו שמן טהור במקדש אלא פך אחד ולא היה בו להדליק אלא יום אחד בלבד והדליקו ממנו נרות המערכה שמונה ימים עד שכתשו זיתים והוציאו שמן טהור:
When klal Yisroel gained control over their enemies and destroyed them on the 25th of kislev and then found the jar of oil. The Misna Brurah writes that the word channuka itself means Chanu= they finished and rested Chuf hei=on the 25th.
It seems construction of the Second Temple commenced on 25 Kislev as evidenced in Chaggai 2 (particularly verses 10, 15, and 18). See these articles by Rav Yoel Bin-Nun (link, long and very thorough) and Menachem Leibtag (link, more succinct; recommended) for more discussion on the significance of this date and the connection of the prophecies there and in Zechariah to Chanukkah.
Edit: Some highlights of the articles (and I do hope you read them inside) that directly address the question:
There is also what to discuss about the prophecies of Zacharia from the same time period as Chaggai (which is the Haftarah for Chanukkah) which center more around his visions of the Menorah, but I'll leave that for the interested reader to find himself.
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Every historical (e.g. I and II Maccabees and Josephus) and traditional (e.g. the Talmud and Megillas Taanis) source that discusses the matter agrees that the 25th of Kislev was the day that the Hasmoneans, having succeeded in driving the Greek forces out of Jerusalem, reinauguarated the Temple service.
As II Maccabees states (10:1-5):
Rabbinic sources add that the miracle of the oil took place at the same time.