The correct translation is "Fata mea învață mult". However, you could force a phrase like "Fata mea învață multe" with a slightly different meaning.

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"Fata mea învață mult". Here, "mult" is an adverb. This meaning is closer to "a lot".

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"Fata mea învață multe". Here, "multe" is an adjective, declined for feminine plural. By that it implies that there should be another feminine plural word coming. So the full sentence could be "Fata mea învață multe lucruri" (My girl learns many things / a lot of stuff). Since "lucruri" (things/stuff) is already hiding what exactly she's learning, the shorter version doesn't detract from the meaning. If anything, it actually adds meaning, notably it is now clearer that the speaker himself doesn't know what their daughter is actually learning. It can be subtle, but the shorter version is now cast behind a thin veil of irony. See an example in English here (Youtube video, only the first 10s from the mark).

Note that in the shorter version, since "multe" is now representing by itself the clause "multe lucruri", it has to take the role of noun, so it is "substantivat" (see dictionary here).

Another point, because I've seen other learners make the same mistake in other sentences.

From a grammatical point of view, you can use both "mult" and "multe" because they don't upset any other part of the sentence.

• The subject "Fata" is singular and it is reflected in the verb "învață" (III pers sing).
• "Fata" is fem. sing. hence the possesive pron "mea" (mine) reflects that.
• But for "mult", "multe", nothing else from the phrase forces it to be one way or the other.

You can perfectly well learn one thing, or many things. The verb is the same. Because the verb depends on the subject, not on the direct object.