I was in a sort of similar situation last summer when I had to live with my mom for a month before I moved across the country. My mom is naturally really thin. She has been working nights for 35 years and pretty much subsists on coffee and cigarettes. She'll eat one huge meal a day around 5pm and that's it. Usually it's brown rice, spinach and black beans -- so not really unhealthy, just very basic and unvaried. Since she lives alone, she doesn't really like cooking for herself (though she is actually a great cook) and she feels like eating a healthy, varied diet is a luxury she can't afford/doesn't deserve.
My situation might be a little different from yours, but the same principles might apply. When I moved in, I tried to motivate her to "rediscover" food. She hates grocery shopping, so she'd give me cash and I'd go get whatever I thought we should eat. I took her to the farmer's market every few days and we picked out the most beautiful produce, locally baked breads, some really memorable honey, etc. We cooked, we baked, we ate something different almost every day -- It was an amazing experience! The best thing was, she became really interested in the various qualities of different foods -- like what flavors are unique to chard versus kale and so forth. The change has been semi-permanent even though I moved away.
So, my advice is this -- don't underestimate your family's ability to take interest in the way you eat and, perhaps, adopt some of your habits. I second the advice to do some of the shopping and cooking, but I'd say take it further than that. Talk to the fam about how healthy eating makes you feel, what qualities of certain foods or food combinations are interesting or exciting. Stick with your plan, be positive and avoid being critical. Really try to be an advocate of healthy eating and appeal to their senses, because that's what eating really is to most people -- a sensory experience.