enough stamina so I can run a mile (or thereabouts) without any significant stopping.
Walk, then run. You can't just drink a protein shake and magically be able to run... But I'd say don't cut out carbs. Eat from the food pyramid (srsly), and jog every day. Stretch, take breaks, whatever you need to do in between. Eventually, you'll be able to run that mile no prob. I used to run two miles nonstop, and that was while I was a heavy smoker (packaday). I just kept tricking myself, saying "ok, I'll run to that spot, then I'll start jogging instead..." but when I actually got to that spot, I'd keep running and think "Nah, man, we're doin' so good, just keep running. I'll stop there." An endless cycle of fooling myself... Totally worked for me. Might not for you, I dunno.
Oh, and can you put a personal trainer in the budget? If not, just find a friend to run with. It's better that way, because there's no way for you to go "oh, no, I don't feel like it today."
And reward yourself.