What's the best course of treatment for tendonitis of the ankle? How much time should it take to heal?

Depends. There are quite a few tendons that surround the ankle, tendonitis at back of the ankle is different than tedonitis over the inner aspect and that on the outer aspect. Best course of treatment is based on evaluation of the tendon(s) and the underlying cause and time taken depends on these as well.
Immobilization. True tendonitis should be immobilized. By restricting motion, tendonitis will improve. Nsaids can be helpful as well.