Actually your weight is pretty much right where you should be for your height and frame. Being concerned about health is very good but don't be overly concerned about numbers on the scale. Most of us aren't built to be 110 lbs and a size 2 and that's ok.
Rather than telling your sister that she's gaining weight and all that you could just lead by example. Invite her along on a walk, a bike ride, swim, playing frisbee, sleding, skating, or whatever else you enjoy doing that's physical. It doesn't have to be a sport, I get that you and she aren't into sports but there are lots of other things to do that are active.
Try to get the whole family involved in activities together. It's not only good for everyone's health but it's good for building your relationships. You could also offer to cook dinner now and then and prepare something healthy and delicious.
Don't get to carried away tho. To many times people try to take bacon eaters to tofu eaters and it just doesn't work. In fact it can turn them off of healthy food completely.
Finally both you and and your sister should watch the Dove Real Beauty videos and remember that you don't have to be a size 2 stick figure to be beautiful. Be healthy and love yourself for who you are!
https://www.youtube.com/watch?v=PPjeelgG9Cg
https://www.youtube.com/watch?v=_3agBWqGfRo
https://www.youtube.com/watch?v=Q5qZedMTkkE