Ambitious, you're female, correct? I would find it hard to believe that a guy who is 5'6 would want to weigh about 120 pounds. But if you are female then it makes more sense. Anyways it looks like you're just thinking about working out and don't really plan on dieting. If that's the case then it's not the best way to go. You can lose weight by working out and sticking to your normal eating routine if you never ate that much to begin with, but it'll be a much slower route and maybe you won't be satisfied with your results.
I'm not expert, I'm pretty much still learning like most people. But my suggestion is for you to diet along with exercising. I would also suggest that you should lift weights as well, lets just say 2-3 times a week at least. It's not really for the sake of gaining muscle, but just to help tone your body and burn more calories besides cardio work outs. Your diet should probably be around 1700 calories that should consist of nutritious food like chicken breast, vegetables, fruits, and so on. Try to get a good amount of protein if possible, at least 100g(most would say the same amount as your body weight though).
Anyways, I hope that helps a little bit. Good luck with your goal. As for how long it will take, with my suggestion maybe 8-10 weeks. You can lose it quicker, but the results(basically how you look) may not be adequate to your liking and the weight may come back quickly depending if you go back to your normal routing depending on how you lost the weight. But with a slow grind, it'll be better in the long term and give you the body you may desire more.