The article quoted by Dr. Balaji Viswanathan (பாலாஜி விஸ்வநாதன்) - What If Gandhi Had Lived On? was a tantalizing read. I will summarize the article which has historical facts on Gandhi's plan post January 1948 and try to extend it based on my imagination.

• The Last Fast: From 13 - 18 Jan 1948, Gandhi had undertaken a six day fast. The fast was to pressure the Indian government to pay Rs 55 Crore (INR 550 million) to Pakistan as agreed upon earlier (the money was paid to India by the British earlier). The Indian government was withholding this money since hostilities had broken out in Kashmir. The fast took everyone including Patel, Nehru and Jinnah by surprise. After there were sufficient written promises made, Gandhi broke his fast. This event made him earn tremendous goodwill with the establishment in Pakistan as well as with the common man.

Gandhi on Jan 22 at a prayer meeting after ending the fast on 18th Jan

• The Trip that never was: It was agreed on January 27 that Gandhi would travel to Lahore on February 8 or 9. He was to be offered protection by the Pakistan Army Police. Jinnah was looking forward to the visit. There was no ambiguity around what Gandhi was going to say. He had put it out upfront that he was planning to appeal to the Muslims to protect the minorities in Pakistan.

• The Letter: A letter from a veteran freedom fighter Konda Venkatappaya stating that Congress legislators were "making money by the use of influence" and "obstructing justice in the criminal courts". Gandhi responded to the letter. This shows that he was very much aware of the unintended consequences of hard-fought independence.

Gandhi was 78 at the time of his death. To answer this question, I am making an assumption that he might have lived another five years (Aged 83 in first half of 1953).

The Agenda:

• India - Pakistan Relationship : This would have been the top most item on Gandhi's agenda. Gandhi was a realist. He probably knew that partition could not be undone. The next best thing would now be to have two countries exist in harmony.
• Dissolution of Congress : Gandhi had been very clear and vocal in his demand that the Congress as a political entity should be dissolved. He wanted it to be a social organization that addressed problems of illiteracy, ill-health, unemployment, untouchability and communal intolerance in every village in India. He was sure that parties across the spectrum from the Left to the Right would spring up once the Congress as a political entity was gone.
• Gandhian Economics : Nehru was the "Commanding Heights" Soviet style central planning guy, Gandhi wanted each village to be self-sufficient. He would have tussled with Nehru ideologically and focused on making India self-sufficient differently.

The Years Ahead:

1948 / 1949:

• The "First Kashmir War" was fought from October 1947 to January 1949. The brunt of the fight took place after Gandhi's death. With Gandhi traveling to Pakistan in Feb 1948, it would have been impossible to continue the fight. A temporary ceasefire would have been one of his conditions during the travel.
• Gandhi was a strategist. After a whirlwind tour of Pakistan in Feb - March 1948, Gandhi would have ensured that Jinnah / Liaqat and Nehru / Patel come to the table to discuss and resolve Kashmir. He would have appealed to the people of the countries to convince their leaders to talk. And he would have pressured them behind the scenes as well. And who better to broker the deal than Gandhi himself? Given that Gandhi had proposed Jinnah as Prime Minister of India in April 1947 in a bid to keep the country united, he would have had tremendous pull on the Pakistani side as well. He would have also had Khan Abdul Ghaffar Khan (a.k.a Frontier Gandhi) for flank support as well.
• If we were lucky, the Kashmir issue would have been doused in 1948 before Jinnah's death in September 1948. I don't want to predict what could have happened (no controversies here). But the issue would have been resolved in some way. And it was important for Gandhi to live for at least 5 more years to cement the deal and make sure it didn't crack at the seams.
• Given that Sardar Patel was integrating other kingdoms into the Indian union wielding both a pen and a gun, Gandhi might have had something to say about that. It is likely that Patel was too quick with Hyderabad and Travancore, while Gandhi was still focused on the frontiers (Punjab / Kashmir).

Nizam of Hyderabad meeting the Iron man before the instrument of surrender.

1950:

• Pacing Himself: Gandhi paces himself. He tries to deal with social issues such as illiteracy, untouchability, etc which were plaguing India. He would have made several visits to the hinterlands and energized the country in general.
• Ambedkar and the Constitution: With the Indian constitution being written by B.R. Ambedkar, there would have been several thought provoking letters and correspondences between the two. While I will not attempt to reconcile Gandhi and Ambedkar (this is an impossible task - both were lawyers who had a very different take on dealing with India's caste system problem) - I am pretty sure that Gandhi's influence on the Constitution would have been present.
• Note: I am at a loss of words here because I do not know the Indian constitution deep enough to point out specific areas where Gandhi might have contributed to or changed significantly. Someone else can leave comments or write an answer that focuses on his impact on the constitution if he had lived.

1950 / 1951:

• Once India was declared a democratic republic in 1950, the country was gearing up towards the general elections of 1951 / 1952. Gandhi would have waited for this. He would have made public appearances and stated that he wanted the Congress dissolved. Imagine Nehru traveling around the country asking for votes for the Indian National Congress and Gandhi traveling around asking for the Congress to be dissolved. It would have been an impossible situation.
• I suspect Gandhi would have had his way and the Congress would have split into two large ideological groups (Center-Left and Center-Right). Both would have had the word "Congress" in their names anyway. Center-Left Congress led by Nehru / Shastri / Kamaraj and Center-Right by Rajaji / Morarji Desai / Syama Prasad Mukherjee. (Note: Patel was dead by this time). Read: Beyond Nehru, Patel: 7 right wing leaders in India who did not get their due at the right time. All other parties would have joined one of these as "coalition" partners after the election. This would have paved the way for reducing the congress legacy and ultimately preventing the Emergency in 1975.
• Knowing Gandhi, he would not have sided towards one party. The effect of the Indian National Congress being the primary party of freedom fighters would have been neutralized. Political parties would have had more economic agendas, social agendas, etc. This would have been a great thing - and Gandhi would have thrown his weight behind the "issues" rather than behind parties. He would have also been a conscience keeper and would have exhorted one and all play fair in elections.

1952 / 1953:

• India now has a new forward looking constitution, a new government and a clear handle on how to handle social issues for the next decade (1952 - 1962). It has friendly relationship with its neighbor Pakistan. There is trust in the air and hope for the country.
• Gandhi is now old and ailing and there is a lot of news around his sicknesses. He is unable to travel very much and a lot of world leaders - ranging from Harry S Truman from the US to those from newly form countries visit him / call him / write to him. He appeals to Truman to stop fighting in Korea. (Who knows what effect this might have had!)
• After being invited, Gandhi refuses to travel to the UK for coronation of Queen Elizabeth in 1952 - but sends his wishes from his Sabarmati Ashram and appeals for peace all over the world
• Being the saint that he is, he now knows that his days are numbered. He continues to talk about how India and the world should move forward and become more peaceful and tolerant in their endeavors.

Eventually, he dies in 1953 - and pretty much has the same funeral that he had in 1948. But only this time, people weren't angry. They knew the time would come and had come.