In terms of Philosophy, how can I make the following argument into a valid and sound argument.
"A closely related objection to utilitarianism charges that utilitarian thinking cannot explain why we should respect people's rights. The American philosopher James Rachels gave the example of a Peeping Tom who secretly took pictures of his undressed neighbor Ms. York:
Suppose that he does this without ever being detected and that he uses the photogrpahs entirely for his own amusement, without showing them to anyone. In these circumstances it is clear that the only consequences of his acions is an increase in his own happiness. No one else, including Ms. York, is caused any unhappiness at all. How, then, could utilitarianism deny that the Peeping Tom's actions are right? This example shows that utilitarianism thinking is not fully compatible with our thinking about moral rights. Most of us would agree that everybody has the right to privacy and that the Peeping Tom has violated Ms. York's right to privacy. Thus, for those who take moral rights to be a central element in their moral thinking, utilitarianism seems uncceptable."
How can a valid and sound arguement be reconstructed from the arguement of the "Peeping Tom"
The most basic way to understand utilitarian philsophy is its aim of achieving the greatest good for the greatest number. You've really already answered your own question here, as you say "Most of us would agree that everybody has the right to privacy and that the Peeping Tom has violated Ms. York's right to privacy."
The greatest good then, would be maintaining that privacy for all. If this peeping tom is allowed, what then is to stop ANYONE from violating privacy rights, whether or not the "offended" party "knows"? It has been deemed by the majority that privacy is important. Therefore, the greatest number of people benefit from this standard being maintained. The "Peeping Tom's" argument that it brings HIM greater happiness is not the concern.