King would have spent the rest of the 1960s and the early 1970s fighting for economic equality and justice. He would have continued to oppose the Vietnam war. Because of this some of his close aides in the SCLC would likely either leave the SCLC or have a sort of hostile takeover the organization and remove King as its leader. With Nixon becoming president, Nixon would have viewed MLK as another liberal opponent and would have had him on his enemies list. Nixon would have in public supported some civil rights issues MLK supported, maybe sign some heavily watered down civil rights bills. But pretty much ignore King and the issues MLK would fight for. MLK might as even go as far as to publicly endorse a presidential nominee for the first time in 1972 by endorsing George McGovern (McGovern would still lose).
By the mid-1970s he would have supported the efforts for American businesses to divest from the apartheid government in South Africa. Would have certainly criticized Nixon over Watergate. With the recessions and high inflation of the 1970s, MLK's economic justice movement might gain some traction with more Americans. By this time he would have started to experience health problems. His family, doctors, and friends could have convinced him to stop smoking, eat healthier, and exercise more. This would have coincided with growing popularity of jogging in the mid-late 1970s. Maybe somebody like former football star Jim Brown could have convinced him to try to live healthier, or at least stop smoking. The deaths of people like Mahalia Jackson (1972), Lyndon Johnson (1973), and Fannie Lou Hamer (1977) might have convinced him to take better care of himself. King would have been a supporter of the women's rights movement and the ERA in the 1970s.
By the late 1970s he would have lobbied Jimmy Carter to push for more progressive economic policies with some success in some areas. King likely would have supported the candidacies of many black candidates for congress. Many of which would have been younger members of the civil rights movement who marched along side of him in the 1960s. King would have lobbied the Carter administration to support sanctions on the apartheid government in South Africa, especially after the Soweto uprising and the death of Steve Biko. He would have found a kindred spirit in Desmond Tutu. President Carter would have given Dr. King the Presidential Medal of Freedom in 1980.
By the 1980s he would have been more outspoken about apartheid in South Africa. He would have visited the country and maybe have a meeting with P.W. Botha. He would have tried to convince the ANC to go back to practicing non-violence to end apartheid in South Africa. MLK would have been a vocal critic of the Reagan administration. Opposing Reagan's cuts to domestic anti-poverty programs. Dr. King would have been troubled with rising gang violence and the growing crack epidemic in the African-American community in the 1980s. He would have been a critic of the American justice system and the high incarceration rate in the black community. Though I think if MLK had lived these problems would not have been as bad as they were/are in OTL. It still would have been a major problem. MLK would have called upon younger African-Americans to become more involved in their communities and continue the movement to empower the black community in the industries of government, business, education, finance, journalism, and law.
By the mid-1980s he likely would have had to slow down even more. If he were still the head of the SCLC, he would have stepped down by this point. Other African-American leaders would have taken over the spotlight, especially if Jesse Jackson is still a national figure ITTL and runs for president in 1984. MLK would have supported famine relief in Ethiopia and other third world countries. He would have been a supporter of AIDS research and for the government to educate the public about the disease. By the late 1980s because of his declining health his public appearances would have decreased even more. Making some talk show appearances on shows like Donahue, Larry King Live, and The Tonight Show. He would have spent more time with his family and friends in his home in Georgia. Making a few sporadic speeches at events involving civil rights, calling for the release of Nelson Mandela, and ending poverty and homelessness. He likely would have made some money writing his autobiography and other books about his beliefs and philosophies.
By the early 1990s MLK would have been rarely seen in public, probably have a debilitating heart attack or stroke. After Mandela's release from prison he would have visited Dr. King in Georgia when Mandela visited America. Taking an iconic picture with him that would be seen as a link between the American Civil Rights Movement and the South African Anti-Apartheid movement. During the 1992 L.A. Riots MLK would have called for calm and peace, but would have been worried about the future of the young black America born after the height of the Civil Rights Movement he led. His last public appearance would have been a White House visit at the invitation of the newly elected president Bill Clinton in 1993. King's health would have continued to decline and he dies in early 1995 at the age of 66 of heart disease.
Getting him to live to still be alive today is not impossible, improbable, but not impossible. But he certainly would have had to make some lifestyle changes in the 1970s in order to add years to his 60 year old heart. Some believe that brother (A.D. King) died in 1969 after having a heart attack and drowned in a pool. MLK's nephews (A.D. King's children) died of heart attacks at young ages. His father died of a heart attack (though he lived into his 80s). So heart disease appears to have been a problem in MLK's family. So his declining health combined with a succession of conservative presidents through the 1970s and 1980s, these things would have limited Dr. King's anti-poverty efforts.