***outline***
I've just come to Chicago from Japan 1 week before and now working
with ten local staffs. I can do almost everything regarding daily
communication and business meetings. Furthermore, I want to imrove my
English skill especially pertaining to speaking flufluency and
grammatical accuracy. My present TOEIC score is 940.

***requirement***
I am looking for the best school or teacher to let me achieve my
purpose in Chicago. If its quality is good, I don't concern about its
fee. Moreover, it is favourable to be near downtown.

If cost is not a factor, I would recommend a personal tutor every time
and I suggest that you advertise in a local paper and see who pops up.

Dear Yasu1965
  I'd like to suggest you contact The Japan Area Language Teachers Association 
(JALT) to see if any of their members are currently living in the Chicago area.
JALT is an association of teachers of English (and perhaps other
foreign languages) in Japan. It has a long history of excellence. 
JALT members tend to be
extremely professional and usually have lived in Japan for many years,
teaching English.  The JALT website has a forum where you could post
questions. The link is
http://forums.jalt.org/
   I think JALT members will have had more experience teaching
business people than other teachers of English as  a second language. 
Also, they will have more access to materials specifically designed
for Japanese students of English.
   Best wishes!
AM

To clarify my previous posting... you might use the JALT forum to post
a message asking if any JALT members are currently living in the
Chicago area.
 Sorry about that!

Since you are really quite fluent in English, I expect that what would
help you most would be tutoring or a course in speech or elocution,
perhaps one of the sites linked here:
http://il.allpages.com/chicago/education/additional-educational-opportunities/special-interest-schools-or-instruction/public-speaking-speech-schools/

You are quite right to start immediately to improve your accent,
before constant English speaking ingrains traces of your Japanese
accent and maybe speech patterns, making it later more difficult to
correct them.

I expect that most of your friends and colleagues will be happy to
help you eliminate grammatical errors if you ask them to do so.  You
may feel uncomfortable about doing this if you are senior to them, but
they will appreciate your own recognition that you know you may be
making a few mistakes and be pleased to help you.

Again, if you wait too long, they will stop hearing them and just
accept that that is the way you talk, and again, the mistakes  - bad
habits -  will be more difficult to change.

I speak from the experience of decades living in the country of my
second language.  :-/

Good luck!

Thank you so much for very useful comments regarding my question. As
one of comments pointed out, my typical Japanese accent and ingrained
speech patterns frequently annoy me. I'm going to check and try all
web sites and associations which you commented on as soon as possible.
Again, thank you so much for your kind and very useful comments.