Please check(This is a corrected version)

Since last week, the dustbins near the Basaveshwar Nagar Post Office building have not been cleared and the filth has been on the road.  The dirty smell emanating from the accumalated garbage has become unbearable and is causing great inconvenience to us.  It has caused various diseases to the residents and people are suffering .  Please take proper action against the responsible people and act accordingly to solve the problem as soon as possible.

I hope you will take action in this regard as soon as possible.

Self-Edited

1.   the dustbins near the Basaveshwar Nagar Post Office building have not been cleared and the filth has been on the road.----------the dustbins near the Basaveshwar Nagar Post Office are yet to be cleared and the filth has been left on the road.

2.The dirty smell emanating from the accumalated garbage has become unbearable and is causing great inconvenience to us.  It has caused various diseases to the residents and people are suffering .---------"The dirty smell emanating from the accumalated garbage has become unbearable and is causing severe health hazards to the residents."

2. Please take proper action against the responsible people and act accordingly to solve the problem as soon as possible. -------"take proper action and act accordingly are redundant phrases" . Please take stringent action against the responsible people to resolve the problem right away. 

3. As Ms. Patty said: Asking somebody to do something soon leaves the timing open to interpretation.  Therefore, I have used "right away" in place of "as soon as possible".

4. I hope you will take action in this regard as soon as possible. -------I hope you will take necessary action/steps to save us from this problem or We appreciate your cooperation in this regard.

Are there any other wordy phrases which you would like to point out? I would appreciate your valuable comments.  Thank you for your time and consideration.

 Mr. Jeff, sir version:

"The dirty smell emanating from the accumalated garbage has become unbearable and is causing severe health hazards to the residents." In the previous sentence, you complained about the full trash receptacles and filth on the street. Telling the reader that the smell comes from the garbage is redundant. That source may be inferred from the previous sentence. --> "The dirty smell has become unbearable and is causing severe health hazards to the residents."

I would find a better expression than "dirty smell". Perhaps "foul odor."

"take action against" is asking for the responsible parties to be punished. Do you want the problem fixed? Or, do you want the guilty parties punished? Or both? As your letter stands, you are only asking for punishment. Would you be saisfied if the guilty parties were fired, but the trash remained? In America, we would ask first for the problem to be fixed.

Ms.Patty, Ma'am version

These edits all look very good, Sanjay.  For the final one, I'd probably keep the tone very serious and urgent.  Hoping doesn't get one very far. How about this?

We will appreciate your immediate response to rectify this urgent matter.

Mr. Peter guess version

Sanjay, taking action against responsible people is not what you need to say – responsible people are the good guys in this scenario. In fact, you yourself are one . . . The people responsible (for this) are the ones against whom action should be taken. You could also (optionally) use a pronoun: Please take proper action against the people responsible . . . Please take proper action against those responsible . . .