Once you figure out your diet, then look into HIIT.
40lbs sounds like alot, but you should easily be able to lose that in 10-20 weeks (2-4lbs/week).
1lb is about 3500 calories. So you need to burn 7000 more calories than you eat per week, to be done in 20 weeks.
Great thing is, you may not even need to go to the gym for HIIT. Just get a treadmill, or a stationary bike (or even a gazelle!), and then a set of lightweights.
Squatting would be good for you, as FDG suggested, but think of HIIT like this: it takes 10-15 minutes everyday. So you could easily wake up and do it, without thinking. And you could also easily stick to it, long after you've lost your weight, as maintenance (for you) would be like 5-10 minutes every other day.