You have to check what le mien/la mienne refers to.
In this case it refers to résolution, which is a female noun. Therefore, you need la mienne.
You are not looking for a “he” or “she”.
You need the grammatical gender (and number) of the noun/possession. The possessor’s identity (but not gender) - my, or your, or his/her/its determines the m- , t- or s-. The rest is determined by the grammatical gender and (grammatical) number, of the possession/s. As Chris notes, “sa résolution”.
The same (mostly) applies when using the possessive adjectives, and because of “sa bonne” you can identify résolution as feminine if you didn’t already know :
Mon, ma, mes; ton, ta, tes; son, sa, ses = my; your; his / her in French (French possessive adjectives)%252Fsearch%253Fs%253DPossessives%2526model%253Dcompetency
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