The first one is correct. That tense is called the present perfect. Here's the conjugation: http://www.verb2verbe.com/conjugation/english-verb/write.aspx

Both are wrong. It's been a long time since I wrote you a letter.

"It's been a long time since I've written you a letter." sounds the best to me, if I had to choose. However, I think this would work too: "It's been a long time since I wrote you a letter." Note that "I've wrote" has become "I wrote".

You've never written me a letter. If you had written, then I would have written back to answer the letter you wrote. What happened to the apostrophe? You should use one in every contraction you write.