It is not "absolutely necessary", but using it is more idiomatic.

Note that we’d rather say :

Ce n’est pas le genre de fille à passer des heures devant son miroir.

or,

Ce n’est pas la fille qui passe des heures devant son miroir.

and in spoken French:

C’est pas…

Most idiomatic, and pretty neutral in terms of formality:

Elle n’est pas du genre à passer des heures devant un miroir.

Remove “n’” in casual spoken French.

In French, if the category (here “fille”) isn’t really necessary, we tend to omit it. “Être du genre à …” is an almost word-for-word translation of “to be the sort who …”, and in French that’s usually the most natural phrasing.

In informal, spoken French, you can also say “être le genre à” or even “être genre à”. I’d categorize it more as the omission of a word than an instance of some more general grammatical construct.

In informal French, you can start with “C’est pas”:

C’est pas le genre à passer des heures devant un miroir.

(Or “Ce n’est pas …” which makes it less informal.) Here you can’t use “du genre à” because that would make the pronoun ce look like it’s referring to a thing, not a person. In “C’est pas le genre à …”, the pronoun ce can be thought of as referring to “genre”, which is not a person and therefore ok. The pronoun ce can refer to a person in the construct “c’est” (plural: “ce sont”), but there has to be something in the sentence that refers directly to a person. For example, this is possible (and without any awkwardness regarding ce referring to a person). It’s not wrong, I just find it too wordy and would use “du genre à” as above instead.

Ce n’est pas le genre de fille à passer des heures devant un miroir.

This is also possible, and a little formal, but again I find it a little too wordy:

Elle n’est pas le genre de fille qui passe des heures devant un miroir.

Regarding the article in front of “miroir”, in French, it’s a lot more natural to use an indefinite article, because it isn’t referring to any specific mirror. If referring to the girl’s routine, she presumably does it at home or with a portable mirror, so “devant son miroir” is also possible, but I find it less natural. It’s also possible to consider “mirror” to be some kind of universal concept and use a definite article “devant le miroir”, but I find this definitely unnatural: it’s more of a figure of speech than normal speech.