For the sake of having a visible case change, let's replace "Linda" with "Bob":
Bob changed his attitude.
1) That did not result in his starting to make attempts.
2) That did not result in him starting to make attempts.
Under a traditional analysis, gerunds and participles form phrases, not clauses. They do not have subjects. This is supported by the observation that "he starting to make attempts" is ungrammatical.
Above, we have two grammatically sound options. In the first, the simple object of the preposition "in" is "starting". The gerund phrase "starting to make attempts" is modified by the attributive genitive pronoun "his". In the second, the simple object is "him". The participial phrase "starting to make attempts" modifies the objective pronoun.
The question is which simple object, if either, is more sensible. Which better represents this particular lack of result: the action associated with the man, or the man in that condition?
To my sensibilities and in the absence of further context, the lack of action is more relevant. This author made the same choice that I would be likely to make.
The simplest thing to do is to assume that the original is as it should be, that it expresses the author's intent, that the lack of action has the greater relevance in the author's mind. The other option is just as valid grammatically, but it places the emphasis on a different constituent.