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:
On the day before the building of the Second Temple was to begin (24 Kislev), Chaggai prophecies to the people, encouraging them to go ahead and rebuild the Temple and promising prosperity and victory over its enemies in its days.
Many years later, on 15 Kislev various idols were erected in the Temple by the Greeks, but sacrifices were not offered on them until 25 Kislev (I Maccabees 1:54-59) which is the day the Temple began to be built. Presumably this date was chosen (by delaying the sacrifices) on purpose by the Hellenist Jews who remembered Chaggai and his prophecies.
3 years later the Jews reconquer the Temple clean the whole thing out (including removing pagan idols on 23 Cheshvan and 3 Kislev according to Megillat Taanit) and rebuild many of the vessels (see 1 Maccabees 4:41-59). Finally on 25 Kislev, the same day it was originally defiled and the anniversary of Chaggai's prophecy that the Jews would succeed and triumph over their enemies (Chaggai 2:22), the Maccabees finally rededicate the Temple in a festive 8 day party. (Note Rashi to Chaggai 2:6 already connects his prophecies to the Maccabees.)
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.
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):
Now Maccabeus and his followers, the Lord leading them on, recovered the temple and the city; and they tore down the altars which had been built in the public square by the foreigners, and also destroyed the sacred precincts. They purified the sanctuary, and made another altar of sacrifice; then, striking fire out of flint, they offered sacrifices, after a lapse of two years, and they burned incense and lighted lamps and set out the bread of the Presence. And when they had done this, they fell prostrate and besought the Lord that they might never again fall into such misfortunes, but that, if they should ever sin, they might be disciplined by Him with forbearance and not be handed over to blasphemous and barbarous nations. It happened that on the same day on which the sanctuary had been profaned by the foreigners, the purification of the sanctuary took place, that is, on the twenty-fifth day of the same month, which was Kislev. And they celebrated it for eight days with rejoicing, in the manner of the feast of booths, remembering how not long before, during the feast of booths, they had been wandering in the mountains and caves like wild animals.
Rabbinic sources add that the miracle of the oil took place at the same time.