Broken off T posts suck. I lost a brand new Firestone 13.6x28 tractor tire to one, my tire shop got Firestone to cover it as a field hazard.
For general flat repair (aka, not a T post), what I'm about to suggest will sound strange. I didn't believe it at first, but after trying it, it really works. Instead of using Slime, use liquid laundry starch. No, I'm not joking. I don't know how, or why, but it works just as well (if not better) than Slime, and costs less than 1/3 as much! I've put it in three or four riding mower tires, and the steer tire of my utility tractor. It really does work.