In the future, you'll have better luck getting corrections if you post paragraphs under NOTEBOOK instead of ANSWERS.
But since you have relatively few mistakes, I will help you.
Dear Mr.Smith (American English uses a comma after a salutation; British English doesn't)
It's great to recall your teaching us English Speech. Today, I am writing to invite you to be a judge for the upcoming English speech contest. ('quite' isn't normally used in a sentence like this. Look for redundancy and wordiness. There's no need to say something is in the past when you are recalling it. 'I' is always capitalized.)
Compared with other teachers, you have more experience in English speech. It is your effort that motivates students to learn English and help them do better at it. Accordingly, it is necessary for you to be the judge and give comments and advice to the participants. The competition will be held at the Center of Students' Activities at 18:00 on December 21. (It's more concise to say "motivates" rather than "arouses the enthusiasm of". "Advice" is noncountable.)
I am looking forword to your attendance. I am convinced that, with your support, the contest will be successful. Please reply at your earliest convenience. (Avoid beginning sentences with "And" where possible. Avoid an unnecessary passive voice.)