You don't have to extradite the authors if they live in countries in which patent protection is strong and the legal systems are functional enough to impose due process upon them and enforce injunctions/damages. Extradition is typically a last resort, typically for criminal proceedings.
Moreover, an injunction is much more likely to be sought than damages, in which case extradition is unnecessary - all the patentees have to do is get an order from a judge forcing the hosts to take their code and site down.
Many countries don't recognize software patents, and the ones that do have some strict rules about what patents are valid. So, are you sure that this is a slam dunk? You don't seem to have any concept that it might be ruled invalid.
Many countries have strong patent protections and no software patents. If you wanted to say "have software patents" instead of "strong patent protections", it would make a bit more sense.
Moreover, an injunction is much more likely to be sought than damages, in which case extradition is unnecessary - all the patentees have to do is get an order from a judge forcing the hosts to take their code and site down.