You said "Gardasil doesn't prevent cancer per se. It treats a handful of strains of the many viruses that can act as precursors to cervical cancers.
If you immunize against a handful of strains of virus's that are precurosrs to cervical cancer, have you not prevented some mystery number of cancers?
Now, I'm not saying that a girl who is given this vaccine is 100% protected, no one is ever 100% protected from a vaccine - however her odds are generally in the 80-90% range of immunity. Certainly, this is better than nothing.
You make a somewhat compelling case for wanting to take the vaccine, if safe.
You have failed to make the case that it should be mandated by law.
Actually it is around the 70% range since the four strains it prevents are associated with 70% of the cases of cervical cancer. The actual percentage risk improvement could be much lower since by avoiding risk behaviors one can lower one's risk exposure drastically.