I expected "girl" to take the definitive form, "fata" without the accent over the second "a." Why not?
When adjectives and ordinal numbers (like first, second, third etc.) are placed before the noun they determine, the noun has no definite article.
A couple of examples: frumoasa fată (the beautiful girl), primul băiat (the first boy).
Because "a doua" means "the second". Using fata would add an extra "the".