I'd just like to point out that disease isn't going anywhere. Every time we solve a disease, others replace it. Why? Because we don't solve diseases in a vacuum. As our lifespans increase, problems we never considered become common. Things like cancer and alzheimer's and failing hearts and strokes. As we "cure" these issues, our lifespans will increase and we will have new problems (or perhaps worse variations of existing problems).
Humans also have the nifty ability of killing ourselves slowly. Even if it isn't drugs like alcohol (or worse) destroying our bodies one dose at a time, when faced with prosperity we eat way too much or eat things that aren't good for us. Diabetes, kidney failure, the list goes on. If we somehow make kidneys capable of regenerating, wouldn't people just drink more? What about other effects of alcohol use, particularly stretched out over a longer lifespan?
If that wasn't enough, humans have also proven very good at killing each other. Biological weapons and the like are only going to get easier to produce as technology becomes more widely available and the knowledge required becomes more distributed. When we start seeing massive outbreaks of man-made diseases that resist treatment, that will add all kinds of new issues.
Some day, we will hopefully explore the stars. Mathematically speaking, we will run into other planets with life on them. Even if it isn't sentient, it will probably cause disease of one form or another in us. And if humans should settle on a world with a different mix of pretty much anything (oxygen, helium, heavy metals, etc), it will cause health problems we haven't seen before. Additionally, until we can come up with better shielding for spacecraft, explorers running around out there are going to be exposed to charged particles and photons from cosmic rays, solar flares, and all kinds of other things. That extra radiation will be a problem for the medical community.
Say we solve those problems. What about extended exposure caused by one spaceship firing on another, disrupting their anti-radiation shielding? What about psychological disorders developed when people start living to 150 or 200 or 250 years old?
If we manage to implant our minds into machine bodies, we just trade one set of problems for another. Psychological disorders will multiply as people live longer and longer. Memory storage and access times, along with memory degradation, will become an issue. Piracy of memories will be a thing, as will hacking people or entire nations and controlling people's bodies remotely. As a mechanical body can't feel, we would need some method of stimulating sensation; as soon as that exists, people will find ways to abuse it with "mechanical drugs" or whatever. Freed from our bodies, how will we reproduce? Clones, synthesized eggs and sperm? Only lower class "real humans" reproducing while upper class society stagnates with the same people in power for centuries?
If we learn to implant our minds into another human body, we still have to deal with psychological issues. And the ethical problems of taking a body from someone else, or else the means to grow a ready-made replacement somehow. And if widespread and no one ever truly dies, that will create a huge strain on our resources and directly lead to war over those resources. Additionally, the human brain only has so much storage space available, so we'd need to compensate with mechanical devices or genetically-altered brains, either of which will lead to new problems. Or perhaps some means of selecting which memories to take with us to the new body, which could lead to interesting psychological issues.
Naw. Disease isn't going anywhere. It will just change shapes.