In Hebrew, the spelling is חֲנֻכָּה. In English, there are several variant spellings, some of which are licit.
For example, the Ch vs. H alternation. This is an attempt to render the ח. Most pronounce the ח the same as the כ (khaf), such that Ch is certainly acceptable. Others pronounce it as more of a guttural, as it is. One common transcription is an h with a dot below it, and where the dot is unavailable, as a mere h. When non-Hebrew speakers use the heh, I don't mind. But when Hebrew speakers use it, particularly when they are trying to appear more scholarly, I admit I get slightly annoyed. I suppose this is because the heh and het are now identical in transcription, such that the reader cannot distinguish between them, and then nothing is gained over the more natural ch, which is similarly ambiguous with the כ. Nothing is gained, and in fact, something is lost, in that the word is now spelled in a way contrary to how most readers (and often the writer) pronounce it, creating unneccessary confusing. In fact, if they would use ch for chet and maintain kh for khaf, as they often do, one could distinguish between the two. But whatever. It is a scholarly convention.
The h at the end (e.g. Hanukka vs. Hanukkah) is also optional. The ה is not pronounced at the end of the word, since there is no mapik, and thus one might write it in to note the Hebrew spelling, or one might legitimately omit it.
On to the real issue: the n and the k. Sometimes they are doubled. Why should we double a letter? In Hebrew, gemination (=doubling) occurs, and it is noted by a strong dagesh. In general, this gemination occurs together with an unstressed syllable with a short vowel. Short unstressed syllables need to be closed (= of the form consonant vowel consonant) as opposed to open (= of the form consonant vowel). Since gemination of a consonant doubles it, gemination of the consonant following the vowel effectively creates a new consonant which closes the previous syllable.
To take the example of חֲנֻכָּה, the ֻ is a short vowel, as opposed to the long וּ. Therefore, the syllable nu, נֻ, would be an unstressed open short syllable (with kha being the next open syllable). This is unacceptable because open short syllables must be closed. Therefore, the כ is doubled by placing a strong dagesh inside, such that the word is chanukka, with a doubled k. The first k closes the short unstressed syllable nuk, while the second k begins the next syllable ka.
(Note that a strong dagesh can appear in any letter except a gutteral (and resh, except that there are rare occurences of dagesh in resh). In this case, it appears in a kaf, in which case we might confuse it with a weak dagesh. A weak dagesh appears in the letters bgd kpt, or possible bgd kprt. This dagesh just changes it from the fricative to the plosive. (On bgd kpt, the strong dagesh also changes it from fricative to plosive, in addition to the usual gemination.) However, the weak dagesh only occurs at the start of a word or after a resting shva - basically, at the beginning of a syllable that is not preceded by an open syllable. And this is not the case here, and so it is a strong dagesh.)
Thus, the doubling of the k in English is licit. I would say it is suggested, though perhaps someone might successfully argue it optional.
However, there is really no excuse for doubling the n. What precedes the nun is a chataf patach, which is a type of shva and certainly would not prompt gemination of the nun. And we see in the Hebrew spelling that there is no strong dagesh in the nun. With no gemination in Hebrew, gemination of n does not seem licit.