First, let's get into what other types of exercises you can do: chair dips, squats, wide fly push ups, pull ups, chin ups, military press, diamond push ups, decline push ups, lunges, and any sort of kettle bell exercise. The exercises you choose are up to you, but I think your best bet given your goals are: chair dips, wide fly push up, decline push up, and chin ups. This will give you a good amount of chest, shoulder, bicep, and tricep work, as these seem to be your target areas.
Second, you need to reevaluate your set/rep count. Instead of doing 5 sets to failure, do 3 sets of X reps for each exercise, where X is a reasonable number of reps given your current strength. Every workout, you will then increase the number of reps for each exercise by 1. Repeat this, deload when necessary, and you will see improvement.
You should also consider "greasing the groove," which is summed up succinctly with the formula Specificity + Frequent Practice = Success. In other words, the more you do something, the better you get at that specific something. Start doing a handful of pull-ups (nothing extreme, maybe 3-5) every time you use the bathroom, or leave your house, or watch a movie. Over time, you will begin to see amazing improvements in your ability to do that exercise.
I'm neglecting the lower body, which I know is not that great, but I'm
trying to focus on my upper body. I want to be stronger, be more toned
and shave off excess fat on my stomach.
No, it is not great, but it is not awful either. With the minimal amount of exercise you are doing, you aren't really in danger of looking like The Hulk with Barbie legs, nor are you in danger of creating any serious muscle imbalances, save for those that present themselves by exerting more than your lower body can handle (i.e. lifting a heavy box). Once you get your life straightened out and more time on your hands, I encourage you to take on a full body strength program (Starting Strength and Stronglifts are the favorites of this particular site, but there are many more out there).
Your diet is pretty much rock solid. That said, I really recommend counting calories for a week, just so you have a ball park idea of what you really take in every day. Knowledge is power.
Your goal for fat loss is contradictory with your goal to increase strength. Building muscle requires a caloric excess to fuel muscle growth whereas burning fat requires a caloric deficit to fuel fat cell loss, so you can see how these two goals compete against one another. You can try to do both at the same time, but won't see optimal results in either, or you can instead do cycles of bulking/cutting where you focus on strength training for a period of time (3-6 weeks) then switch focus to cutting fat through a leaner diet and increased cardio.