Start using a standing desk. Being sedentary all day is slowly killing you, and I think you will notice a change in your energy levels. I've never had a dask/sitting job, and I'm always on my feet, and I love the gym.
I didn't used to love the gym, but once you get used to that amount of work, your body adapts and it becomes a challenge that you can handle, and feel motivated by.
YOU are your own worst enemy, with feelings of "OMG why are those people looking at me, are they judging me? Am I doing this wrong?" People in the gym tend to stare without knowing it because there's really nowhere else to look between sets.
Stop thinking in terms of a minimum. You'll never make any progress that way. Instead think of it in terms of a commitment. Say to yourself, "Alright...this week I can commit 4 hours to the gym." Really that's not too much, if you break it up into 3 or 4 workouts per week, and if it's close by, change at home, bike there, chain up your bike and get in and get out quick. That way you can bike straight home and eat/shower/relax/whatever. Before you know it, you're seeing results, and all it took was a little willpower. Change the way you think to change your body.
EDITED in response to Loilo:
Look I agree that hacking is a useful tool, but sometimes you just have to put in the work. Good for you Loilo that you can stay as fit as you are without going to the gym. I on the other hand have to deal with a set of circumstances, including being a hardgainer, and low motivation, that means I won't exercise unless I have to. If I don't get out of the house and to the gym, my session for that day is out the window.
I also agree that you should definitely cut out the things that don't work, or are just extra work with no results, but again, doing the minimum out of the gate, is not a good philosophy. I think it's much more efficient to start at a certain level, say 3-4hrs/wk and then scale back after seeing what works. That's how I started out, and I now only go 2-3x/wk, sometimes only once.
If you work out at home, I would suggest having a specific area (if you have the space, I do not, thanks NYC studio apt!) to set aside for working out. I find it easier to adjust my mental state if I "go somwhere else" to engage in exercises. Check this guy's home gym out: