Push Ups and Sit Ups are just fine. You can try pushups with your knees on the floor at first and once you build up strength you can go to the traditional form. When your strength build up even more, move to elevating your feet on a bench (or chair) in order to increase both intensity and mimic the form of an incline bench press.
Sit ups should be in the form of crunches - You can also add knee raises by sitting on a bench and bending your knees up to your chest. This will work your lower abs.
All of this can easily be done from the comfort of your living room - Or in your case, your computer room! (lol).
Rememeber though - Nutrition plays an even larger role in weight loss. All the excercise in the world wont burn off the crap and junk we sometimes eat!