Writing an email to a Japanese teacher

I have been in touch with a Japanese Ikebana teacher for a while now,but since my Japanese isn't that brilliant(not to mention my Keigo skills),I am a bit stuck.First of all,I would like to say to him that I am sorry for the late reply,should I use しばらく連絡をとってなくて申し訳ありません。? Also,I would like to thank him for sharing one of his projects with me and tell him that I think it's wonderful.
And my last question: from what I understood from my teachers,when a Japanese person asks お元気ですか？, I should answer ''Yes I am fine'' or anything like that.Instead,what would be an appropriate answer?
Thank you so much in advance!

It depends on how much of "keigo" you are familiar with (I wouldn't recommend writing something you don't really know), but considering he is your teacher, you could write something like the following.

ご無沙汰しております。(Gobusata shite orimasu.)"Long time no see/no communication" ^ an all-encompassing phrase that says "it's been a while" generally, not specifically apologizing for the lack on YOUR part of communication.

しばらくご連絡さしあげず、申し訳ございませんでした。(Shibaraku go-renraku wo sashiagezu, moushiwake gozaimasen deshita.) I apologize that I have not contacted you for a while.

If you were supposed to "reply" to something specific, a request or a question, then you can use:
お返事が遅くなり、申し訳ございませんでした。(O-henji ga osokunari, moushiwake gozaimasen deshita.) I apologize for the late reply (which I should have sent realier.)


About "sharing one of his projects" could you be a bit more specific? Has he written to you about an exhibition he is planning, or has he sent you photos of an exhibition he's had, etc? I don't want to use the phrase "purojekuto" in katakana :)

In response to "o-genki desu ka," you could say:
おかげさまで、元気にしております。(O-kage sama de, genki ni shite orimasu.)
おかげさまで、元気です。(O-kage sama de, genki desu.)
This says "Thanks to you, I am fine," though he hasn't done anything to help you with your health, this expression is commonly used, also as a way to express your gratitude for thinking of and asking after how you are doing.
You could add:
先生も　お元気でお過ごしのことと思います。(Sensei mo o-genki de o-sugoshi no koto to omoimasu.) I trust that you are doing well too.

See if you feel comfortable with these expressions? :)

You can say "First, excuse my poor Japanese".
For example, 日本語が下手でごめんなさい ?
I think the teacher will be highly pleased to read the letter you write as you can.

Japanese written formula...
"I am sorry for the late reply".
ご無沙汰をいたしまして申し訳ありませんでした。
"Yes I am fine"
おかげさまで、私は元気にしております。
(おかげさまで is very Japanese, but very typical phrase.)

So, excuse my poor English(^-^;)

Thank you very much for your answers,you have been very helpful!I am a bit familiar with these forms of Keigo,but since I rarely use it in e-mails,I am a bit unsure.
Again,thank you very much!