Hi,
Once upon a time, I knew about the British School of Languages, which was doing a good job. They had branches all around. As I do not know how good they are today, I would advise you to do a little research focusing on this.
The British Council focuses on language development. The Delhi Centre, for sure, provides training on language improvement.
This is the link to the Mumbai Centre:
http://www.britishcouncil.org/india/...egional-mumbai
Please see if they can be of help. There must be many quacks who offer instant solutions. But, I guess it takes a while to refine your linguistic skills, and the experts at the British Council will help you establish a strong foundation.
Regarding your sentence, I would refine it and rewrite it as follows:
"Can anyone suggest a good institute in Mumbai where I can improve my English (linguistic skills) within a month?"
Please let us know.
Regards,
Bijoy