In a business letter, after describing a principle of operation and an issue to be resolved or an approach to a problem, one might say "Please proceed accordingly".
In the letter to Mr. Singh seems like it is semi-personal. Based on what some of the Indian English speakers suggest, the phrase "please do the needful" encodes a fair amount of information: Mr. Singh is in a better position to help, and you don't only want him to do something to help you but proceed in a way he's comfortable with, and you'd also like that he feels good about helping you and not "prescribed to". So it is kind of a polite recognition of his status in this particular case.
Now, if we wanted to achieve a similar effect, depending on how well one knew Mr. Singh, in US English we might say (based on an earlier answer)
My cousin would like to learn guitar and I recall you have excellent musical contacts. It would be really great (or "very nice of you", or "really helpful") if you would you connect me with them somehow."
So "somehow" is "the needful", left up to Mr. Singh to devise as he is the expert, and you are recognizing that he's doing you a favor and are appreciative.
In Canada one might say "would you kindly connect me with them?" but in US English this would be maybe a little affected, a bit too formal sounding.