Today Was An Uncomfortable Day.  << title = capitals

오늘은 불편한 날이었다.   Today was an uncomfortable day.
내 핸드폰을 기억하지 않았어요. 깜빡했어요!  I didn't remember my cel phone.  It slipped my mind!
들고나가느 것을 잊어버렸다.
왜 그랬을까?...바보같이...                             Why?  Seems stupid.
그래서 나는 회사에 약 5분정도 늦었다.     So I was 5 minutes late for work. < for work is more natural in  English 네...여기, 이것이 "office" 네가  말했라고 알아요.
나는 결심했다!! 핸드폰을 꼭 챙기겠다고...  < So I decided.... I have to  ~~  (미안해요. 이해할 못했어요.)

I don't understand - pick a mobile phone?  You have one, you just forgot it?
Hmmmm.   I wrote the Korean for a simpler expression.  I hope you don't mind.
^^ 낸시

Today was an uncomfortable day.

Today was an uncomfortable day.
because I don't take out my Mobile phone. I forgot it!
Today was an uncomfortable day because I didn't take my cellphone. I forgot it! OR
Today was an uncomfortable day because I forgot to take my cellphone.
   A because adverbial clause at the beginning or middle of a sentence is separated from the main clause in the sentence by a comma (if at beginning) or 2 commas (if in the middle). Example: Because I was tired, I stayed home. At the end of a sentence (as you are trying to use), the comma is optional (contrary to what SarahCC says). It kind of depends on the length and breath volume (?) of the first part. When you read the sentence outloud, you will put the comma in if you find that you need to take a breath before completing the whole sentence. If the first part is short, then you can probably read the whole sentence in one breadth and therefore omit the comma before the because.
   The purpose of punctuation is for you to able to get a feel for how a sentence should be or would be spoken by the writer (speaker behind the page).
   Note: cellphone is now a word in the MerriamWebster dictionary see http://www.merriam-webster.com/dictionary/cellphone

why did I ?...stupid......
Why did I? ..stupid...   OR
Why did I? Stupid.  OR
Why did I? Stupid...   OR
Why did I? Stupid me.
  Although it wasn't acceptable decades ago (it was called a fragment), a single word or phrase seems to be acceptable these days if used sparingly and effectively.
  The use of ellipses (the 3 dots) is a bit trickier. They usually mean that a word or words have been omitted. You can pick any of the above corrected alternatives. But I think the last one might be best.

So I was late to the office about 5minutes...
So I arrived late at the office by about 5 minutes.  OR
So I got to the office about 5 minutes late.
  Again, usage comes into the picture and overrides grammar. I was late to the office is incorrect, plus you can drop the preposition by. You get to a place late. OR You get to a place late (by) about x minutes. I arrived at the meeting about 5 minutes late.

I have decided!! I'd better pick a Mobile phone..
I have decided [that] I'd better [go back and] pick up [or get] my cellphone.
    Not sure if this is what you mean (go home and get the phone).
  
   Also, the use of mobile can probably be used instead of cellphone, as in I left my mobile at home. I think AU, CA, US, UK English will have probably on or the other as the popular was to talk about your cellular phone. NB. But I kind of like the Korean Konglish expression handphone 한드폰, as people very often have their phone in the hand.

오늘은 불편한 날이었다.
왜냐하면 내가 핸드폰을 갖고 나가지 않았다. 들고나가느 것을 잊어버렸다.
왜 그랬을까?...바보같이...
그래서 나는 회사에 약 5분정도 늦었다.
나는 결심했다!! 핸드폰을 꼭 챙기겠다고...

Today was an uncomfortable day.

Today was an uncomfortable day, [use a comma to connect to the word "because"]
because I don't take out bring my mobile phone.  I forgot it!
Why did I do that?...stupid......
So I was about 5 minutes late to the office about 5minutes...
I have decided!!  I'd better pick a mobile phone.. [This last sentence is a little confusing.]

오늘은 불편한 날이었다.
왜냐하면 내가 핸드폰을 갖고 나가지 않았다. 들고나가느 것을 잊어버렸다.
왜 그랬을까?...바보같이...
그래서 나는 회사에 약 5분정도 늦었다.
나는 결심했다!! 핸드폰을 꼭 챙기겠다고...

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Oh, I hate when I forget my phone!  >.<
Let me know if you have any questions about the correction!^^