I don't think there would be any hope of actually collecting the bits piece by piece, the amount of fuel required to match trajectories alone makes it infeasible. The best 'solution' would be to change the trajectory of the garbage either away from the atmosphere, or into the atmosphere. Unfortunately a lot of it won't completely burn up before it hits the ground, creating a few more problems then just some broken satellites.
I wonder if it would be practical to mitigate ground strikes by steering objects into collisions. I guess if you got the math right you could do the collisions at low orbits and avoid making things worse.
The real problem is the junk is travelling at high speeds, in many different directions. So the only 'safe' way to collect it is to match the speed and direction of each piece of junk, one at a time.
