At less than 12 stone at 5'10" - even if you are a bit fat, you can't be more than 10lbs overweight. Of course you can go to the gym. Four stone overweight fifty-five year old women go to the gym. My grandmother goes to the gym and she is in her eighties.
So no, starving yourself and remaining inactive probably won't work. Eating a diet of fish, milk, lean meat, eggs, rice, potatoes, veg etc. and getting in some good quality exercise will. Getting in exercise will also go a long way to solving your sleeping problem. Trust me, if you try and go for a run a few times a week straight after you wake up and do a few gym sessions during the week, then you will be wanting your bed in the evening and won't be waking up at 4am.
At your weight, even with very gradual weightloss, you could be in pretty good shape in as little as two months.