I don't believe there's technically a grammatical "mistake" in either of the two sentences It's mine nor I admit breaking the window, just some minor faults of conventional punctuation in how you offered them in your question, namely the space you have before the apostrophe with "It 's" and the lack of at least one space after the period/full stop that concludes the first sentence.
But I suspect that you were given this short two-sentence monolog for the purpose of recognizing that the use of the possessive pronoun mine in the first sentence is inappropriate since, as is normally required of English pronouns, it lacks a definite antecedent in this context, antecedent just being the term for a noun coming somewhere before the pronoun that the pronoun is supposed to refer to. Instead, the reference of the first person singular possessive pronoun mine seems to look forward to the second sentence, since the subject of the latter is the first personal singular subjective (or nominative) pronoun I. That is, I believe most native speakers of English would instead say something like It's my fault. I admit to breaking the window, using the noun fault preceded by the correct and appropriate first person singular possessive adjective my (though many people would still--I believe confusingly--also call my a "pronoun").
Another minor point might be that some speakers could insist that the preposition to is required after admit, but I would disagree in pointing out that while this is generally true if what follows is a noun or noun phrase, e.g., I admit to the destruction of the window and I admit to the charge, the preposition is not necessary if the complement of the verb admit is another verb or a verb phrase, in which case having the verb in its -ing form is all that is required as we see with breaking in your example.
Ed M.
04/13/16