There maybe an extensor tendon avulsion or a tear! 2 months is too long to expect any healing by using conservative methods. Again, a physical examination is required to see where the pain is, and where the mechanical failure is, either at the first joint in the finger or the second. Treatment depends on that too.
Surgery is a must in case of a complete tear or an avulsion.
Next Steps
1. physical examination to assess the site of injury
2. surgery in case of a complete tear
3. pain killers in case of pain, on an SOS basis