We have all heard the saying that "Laughter is the best medicine" and some people in the medical community have been sceptical of the healing power of laughter, but that's changing. In fact, doctors have begun to support laughter therapy to complement traditional treatments for injuries and diseases like cancer. From research, it does seem like laughter is the best medicine (in compliment to other traditional medicine).
Psychologists studying the science of happiness believe that practising certain positive behaviours like laughter can bring the pleasure, engagement and meaning that comprise happiness (Jameson, 2008)
Some serving in the military are being trained to use genuine and forced laughter as stress relievers to help prevent and treat conditions like post-traumatic stress disorder (PTSD) that can result from harrowing experiences like combat duty (Burbank, 2006)
According to helpguide.org,
Laughter relaxes the whole body. A good, hearty laugh relieves physical tension and stress, leaving your muscles relaxed for up to 45 minutes after.
Laughter boosts the immune system. Laughter decreases stress hormones and increases immune cells and infection-fighting antibodies, thus improving your resistance to disease. (See also @JohnP's answer in the Health Stack Exchange question Does laughing improve immune system?)
Laughter triggers the release of endorphins, the body’s natural feel-good chemicals. Endorphins promote an overall sense of well-being and can even temporarily relieve pain.
Laughter protects the heart. Laughter improves the function of blood vessels and increases blood flow, which can help protect you against a heart attack and other cardiovascular problems.
Laughter burns calories. OK, so it’s no replacement for going to the gym, but one study found that laughing for 10 to 15 minutes a day can burn about 40 calories—which could be enough to lose three or four pounds over the course of a year.
Laughter lightens anger’s heavy load. Nothing diffuses anger and conflict faster than a shared laugh. Looking at the funny side can put problems into perspective and enable you to move on from confrontations without holding onto bitterness or resentment.
Laughter may even help you to live longer. A study in Norway found that people with a strong sense of humor outlived those who don't laugh as much. The difference was particularly notable for those battling cancer.
Paediatric hospitals also use therapeutic forms of entertainment like clown therapy to help children tolerate the pain, nausea and anxiety associated with chemotherapy and radiation treatment. The 1998 movie "Patch Adams" was based on the real-life story of a doctor who used humour to treat patients while he was still in medical school. Patch Adams and his colleagues at the Gesundheit Institute in Virginia use a combination of clowning, singing and dancing to treat patients through entertainment in conjunction with traditional medicine (Gesundheit Institute).
Other internet sites including How Stuff Works — Science states
Laughter sets off a chain reaction throughout the body that promotes physical and psychological health. Regular and frequent guffawing can both prevent illness and help you get well. Laughter lowers your body's hormone and cortisol levels. (Cortisol is a stress-induced chemical that can lead to heart disease, high blood pressure and excess belly fat.) Laughter also strengthens your immune system because it increases the production of antibodies in your saliva and in your bloodstream to stave off bacteria, viruses and parasites.
Laughter also helps with skin conditions. People suffering from eczema noted improvement in their complexions after consistently watching funny movies. When allergy patients laugh regularly, it reportedly shrinks their welts (Kimata, 2009).
Some cancer treatment centres incorporate laughter therapy sessions into a patient's treatment plan to increase the positive thinking and happy feelings that support the healing process (CTCA, n.d.)
As for laughter curing diabetes, I don't know about curing diabetes, but as How Stuff Works — Science states that laughter reduces cortisol levels which can lead to excess belly fat getting rid of belly fat may lower Type 2 diabetes risk. So diabetes might not be able to be cured by laughter, but laughter can prevent Type 2 Diabetes.
Burbank, L. 2006. The Growing Popularity of Laughter Therapy NPR [Online]
(MP3 Audio) https://ondemand.npr.org/anon.npr-mp3/npr/day/2006/01/20060120_day_11.mp3
CTCA, n.d. Laughter therapy Cancer Treatment Centers of America [Online]
Jameson, M. 2008. C'mon, get happy Los Angeles Times [Online]
Kimata, H. 2009. Viewing a humorous film decreases IgE production by seminal B cells from patients with atopic eczema Journal of Psychosomatic Research 26(2): pp 173–175
DOI: 10.1016/j.jpsychores.2008.06.006 PMID: 19154860