Since you're leaning more towards the coaching/mentorship role rather than direct manager, I suggest making the individual more aware of their accountability. There is an increasing number of programs that help track time and help with productivity, many of which also include smart phone applications.
This can be mandated for that individual or for the team. Although it has a Big Brother ring to it and many on the team may feel targeted in the crossfire towards disciplining one individual, there are ways around this. For instance, the information does not need to be collected and can serve only to help them manage their own time, while strategically speaking to the individual about their work habits.
Suggested programs include a number of extensions and add-ons for browsers (Chrome/Firefox) that track web history and time spent on each domain. Other software like "Toggl" tracks time in general and can be purchased with a team license or free to try.
There is the ethical factor that has not been addressed, as this individual is being paid to perform and adhere to the culture and rules of the organization. If they are wasting their spending time, they are also wasting the company’s time and potentially making an improper earning. Especially in tough economic times like today, many people out of work and I am sure that any one of them would trade anything for their position.
If the matter is immaturity then a more direct approach is needed. I am not suggesting terminating or disciplining the employee but sometimes the hardest lessons are those that need to be learned through experience. These situations call for special skills that can be learned from real life coaches that deal with rough personalities, big egos and high stake situations. Often applying the wrong amount of pressure can backfire and have the individual shut down and feel betrayed. If you are interested in investigating this approach, I suggest looking into resources of teaching or coaching troubled youth or talent.
If they continue to not take their work seriously then there may be something deeper going on than the lack of motivation or ungratefulness; perhaps relationship or family trouble. As a coach investigate these, by befriending from a distance and not getting emotionally attached or personally involved.
If all fails, don't be discouraged. You did your best to wisely inform this individual to shape up and their negative influence spreading to others. It is also potential learning experience for you. Remember to always keep your cool; as a coach you have vested time and effort in this individual, but their eventual success or failure can only be determined by them alone. Best of luck!