Why is there no mention of the miracle of the candles burning for eight nights in Al Hanisim?
Somewhat related to Shalom's answer, the Lubavitcher Rebbe zt"l explains (Likkutei Sichos, vol. 25, pp. 235ff) that the miracle of the oil (a spiritual victory) overshadows the miracle of beating the Greek armies (a physical one) - because the latter was itself part of a larger spiritual war: they tried to make us "forget Your Torah and abandon the decrees of Your Will"; ordered the Jews to "write on the horn of an ox that you have no portion in the G-d of Israel"; etc.
So when we want to mention and describe the miracle of the military victory, we have to omit mention of the miracle of the oil. Conversely, when the Gemara explains מאי חנוכה, it mentions the victory only incidentally.
Al HaNisim was designed to be placed in the blessing of "Modim", in which we give thanks
for our lives which are in Your hands, for our souls which are under Your watch, for Your miracles each day to us, for Your wonders and kindness at each moment, night, morning, and afternoon ...
The focus is on the "everyday" miracles that occur within the bounds of what we call nature. So Al HaNisim likewise focuses on the military victory, which wasn't (blatantly) supernatural.
(This isn't my answer; I heard it from someone contemporary whose name I sadly don't recall.)
A wonderful answer to this question us given in the sefer Tal Oros by Rabbi Yehudah Tuvia Guttentag (Tavyomi). The full text in English can be found here, but the following excerpt contains the essential points:
After the great victory of Yehudah the Maccabee against the many and strong Greek troops and over the multitude of Hellenists, and after the purification of the sanctuary, the rebuilding of the altar, the miracle of the flask of oil and the celebration of the re-inauguration for eight days, Yehudah the Maccabee and his Beis Din, the Great Beis Din that was called at that time “the Beis Din of the Chashmonoim” (see Avodah Zorah 36), they wanted to make a remembrance of these salvations and wonders, and to establish these days of Chanukah as a festival for the generations with Hallel and thanksgiving.
However, on this point there was a confusing problem; on what to substantiate and base this festival? For which remembrance to establish it? If to commemorate the great victory of the few against the many, and the weak against the strong - this would not be in accordance with the rules of the Torah, because a festival with Hallel and thanksgiving can only be fixed for a completely unnatural miracle, as mentioned above. If to commemorate the miracle of the flask of oil that only contained enough oil to light for one day, but by a miracle they managed to light from it for eight days, - this is an ‘inside miracle’, a miracle which occurred only in the sanctuary, in the presence of a handful of Kohanim whose day of service it was, or whose week it was, in a place where it is totally forbidden for non-Kohanim to enter. If in earlier, very righteous generations little importance and value were given to ‘inside’ miracles like these, all the more so in that generation where there were numerous Hellenists. Certainly the general population would not believe in this miracle; they would have no regard for it and would not celebrate any festival in order to remember it.
After great deliberation, Yehudah the Maccabee and his Beis Din came to a decision: they would establish this festival of Chanukah with Hallel and thanksgiving and candle lighting, but without giving a reason and substantiation for establishing this new festival, similar to other decrees and safeguards that were sometimes issued without explanation. Like the gemara Avodah Zorah 30 says, “When they made a decree in Eretz Yisrael they did not reveal the reason for twelve months, perhaps somebody might disagree with the reason, and the people would come to make light of the decree”, and thus undermine the decree.
With this the Beis Din of the Chashmonoim have satisfied everybody. Because those who knew the Torah, who were expert in its laws and knew that a festival with Hallel and thanksgiving is only established for a non-natural miracle, they would trust the testimony of the Kohanim in the Beis Hamikdosh, amongst whom were certainly some of the great Beis Din. They would not doubt even for a moment the miracle of the flask of oil. The general populace, in the absence of an explicit reason, would think that the festival was established as a remembrance of the salvations and victories, and the lighting of candles is to remember the renewed lighting of the menorah in the Beis Hamikdosh. Let them think what they will, as long as they observe the festival with all the its commandments, according to Halachah.
This corresponds to the text of Al Hanissim. There, also, everything is mentioned in a vague style - the wars, the victories, together with the purification of the sanctuary and the lighting of the candles, which is based on the miracle of the flask of oil. But this miracle is not mentioned explicitly, because this prayer is for everybody, and therefore they intentionally concealed this reason of the ‘inside’ miracle because of ignoramuses.