In the inanimate “of which” sense, which's arises out of doubt about whether whose can be used with an inanimate antecedent. In standard English, whose is the possessive of form of both who and which, so whose is appropriate for such antecedents.
In the noun-phrase-final sense, which's is best analyzed not as a single word, but as the last word of a noun phrase, the whole phrase to which 's is appended. In this sense, changing which's to whose is hypercorrection.
which + possessive clitic 's