The first one is fine, although it's very formal. The second one should be "I was quickly convinced of Jack's high level of persistence". "Level of persistence" sounds sounds like something from a video game to me. It might be better to put an adjective before "persistence". 'Professional' should be 'professionally' in the third one. It'd be more natural to move "both personally and professionally" to the end of the sentence. Also, I don't understand what you mean by "the highest loud".

(1) I hope this letter can help you know more about her. (2) I was quickly convinced of Jack’s high level of persistence. (3) I appreciate the importance, both personally and professionally, of her continued education, the highest laud. I think you mean "laud" instead of "loud."

not really. i would write for 1 I hope this letter can provide you with an understanding of her character. or i hope this letter will help you know more about her. 2 I was quickly convinced of jacks high level of persistence. 3 the highest loud? not sure what you are trying to say.