Anglicization from Hebrew
How should one transliterate the well-known Jewish Holiday that usually takes place in December (or late November)?
Hannukah
or
Chanukah
It depends on what the purpose of the transliteration is.
If it's just to be recognizable in writing, then any of the common forms will do. However, if the goal is to signal the correct pronunciation, then "Hanukah" (or "Hanukkah"), while common, fails to capture the gutteral at the beginning of the word and beginning with "ch" is better. I hear lots of people say the word with a soft "h", presumably because of this transliteration; it's clear what they mean but they're mispronouncing the word.
If the purpose is to unambiguously indicate how the word is spelled in Hebrew, which is חֲנֻכָּה, then there are three problems:
The initial letter is one of two gutturals [*]. It's common to render that sound as "ch" (as in German), but that's ambiguous. The academic-style transliterations that I've seen use "ch" for chet (the letter here) and "kh" for khaf (the other possibility).
The final "h" is silent and is sometimes left off ("Chanuka"), but it corresponds to the Hebrew hei so include it.
The kaf, corresponding to the "k" sound, has a dageish in the Hebrew. A single "k" ("chanukah") would properly indicate that sound but would not necesarily make the presence of the dageish clear. A common practice is to double consonants that hava a dageish; another example is the "b" in "Shabbat".
So the most precise transliteration would be "Chanukkah". However, many popular sources including Wikipedia instead render it "Hanukkah".
A final note: since the nun does not have a dageish, doubling the "n" isn't technically correct. Nor does it convey anything about pronunciation. So regardless of the other decisions you make, you should stick with one "n".
[*] Yes yes, there are really five, but only two that make this particular sound. I'm simplifying.
I believe the correct and widely used term is Hanukkah (note that this is slightly different than your version starting with 'H'). Main reason for that is that 'H' replaces the 'Het' (ח) of the Hebrew alphabet, while 'Ch' replaces 'Chaf' (כ) without a emphasis.
Most online sources I reviewed uses the term 'Hanukkah'.
I hope this helps.