What did Machiavelli feel was wrong with Italy of his time, that it needed The Prince he described in his book, The Prince?
When Niccolo Machiavelli wrote "The Prince" in 1513, he felt that there was a great deal wrong with the way Italian governments (of course there was no such thing as the country of Italy at this time) were being run.
The major problem, to Machiavelli, was the fact that Italy was disunited. This gave Spain and France, especially, points of access through which they could interfere in the politics of the Italian states.
A secondary problem, and one which also helped keep Italy disunited, was the amount of temporal (non-religious) power that the Catholic Church had.
Because of this, Machiavelli believed Italian rulers needed to be more ruthless in pursuing their goals. He wrote "The Prince" as a manual for achieving political success by (in part) disregarding notions of morality.
A phrase like "wrong with Italy" implies some common problem or shortcoming of the people of Italy. Machiavelli does not speak about any such problem or shortcoming. Also the book Prince written by him is not directed at solving problem. It is only about gaining and retaining political power. For this Machiavelli covers methods of dealing with political adversaries as well as of ruling the people of a state.
The objective of this book as envisaged by Machiavelli was to help people from princely background with basic abilities to make better use of these assets. This becomes quite evident when we consider the following statement made by Machiavelli, at the beginning of book, dedicating it to Lorenzo Di Piero De Medici:
Take then, your magnificence, this little gift in the spirit in which I send it, wherein, it be diligently read and considered by you, you will learn my extreme desire you should attain that greatness which fortune and your attributes promise.