helping my professor is almost done.

3months ago, I've started to help my professor write his paper as a part-time job.
And 3months later, It will be done!!
Now I'm doing a final (or) finishing work (I don't know how to say it) and then, We're going to fix it.

Although I couldn't make head or tail of this sentence, I though it would be advisable to point out that you should consider PARALLELISM. ( For instance:

 My dad and and mom (both of them are noun).

 beautiful and long (adjective). 

 I sit and look (base form of the verb).

I enjoy singing and dancing (not I enjoy singing and dance).)

YOUR SENTENCE : I'm doing a final (or) finishing work  ---- the correct form: I'm doing .... (and/or) finishing .....  .
Because of this work, I couldn't be using Skype and this web-site.
I can't stand talking with my friends!! haha
Thanks for reading my diary haha

I am helping my professor, and I am  is almost done. 

3 months ago, I've started began to help my professor write his paper as a part-time job.

What do you mean by "write his paper"?

Do you mean that you are helping him correct homeworks or something like that?
And after 3months later, It will be done helping him!!

Now I'm doing a final (or) finish work (I don't know how to say it) (for him) and then, We're going to fix it.  correct the paper.

I can't help you with this sentence, because I don't know what you are trying to say. But to make the sentence better, use final work, and if you are doing it for your teacher, add "for him". And by "fix it" i assume your talking about correcting your paper?

Because of this work, I couldn't be using can't use Skype and this web-site.
I can't stand not talking with to my friends!! haha 

If you don't have "not" in the sentence, you'd be saying that you don't like talking to your friends, which I assume you do.
Thanks for reading my diary haha

I hope you understand my notes ^^ 

Blue = Added words

Grey = Deleted words

Orange = Can't understand what you're meaning

Pink = Notes

I am helping my professor, and I am almost done. 

3 months ago, I began to help my professor write his paper as a part-time job.

And after 3months, I will be done helping him!!

Now I'm doing a final work (for him) and then, We're going to correct the paper.

Because of this work, I can't use Skype and this web-site. 
I can't stand not talking to my friends!! haha 

Thanks for reading my diary haha