Translation:She feels confident enough to speak Dutch.
Good one. Your sentence sounds a bid odd, I would use Ik voel me niet zeker genoeg om Nederlands te spreken or maybe Ik voel me te onzeker om Nederlands te spreken. I wouldn't use anything with zelfvertrouwen (self confidence), it's not wrong, but to me that comes across as a lack of self confidence in gerenal, while the sentence is only about the skill level in Dutch.
Would it work with 'haarzelf' instead of 'zich': 'Zij voelt haarzelf zelfverzekerd genoeg om Nederlands te spreken.' ? If so, would the translation: 'She feels herself sufficiently confident to speak Dutch.' be an acceptable translation? Or is the 'zich' simply lost in translation?
'Haarzelf' can only be used as an object pronoun, and is not used as a reflexive pronoun. What I mean with this, is that 'haarzelf' is the emphasised form of 'haar' (object pronoun), and is used when the subject and the object are different. A reflexive pronoun is used when the subject and the object are the same (so it used when a reflexive verb is used).
In English, 'zich' is not translated in this sentence, because 'voelen' is not a reflexive verb in English (but in this sentence, it is in Dutch!)