It's a harder scenario than you may think. Satellites are built to withstand impacts up to a certain size because the chances of collision with very small debris is almost a certainty for satellites that are going to be in orbit for several years. A BB sized particle could do a lot of damage depending on the vector between the BB and the satellite. Satellites like the Hubble are in low inclination orbits, so the relative velocity between polar orbiting BBs and those satellites would be high. LEO photographic reconnaissance satellites are in a sun-synchronous orbit with an inclination of about 96 degrees to the equator. A BB in a polar orbit (90 degrees inclination) might not have a very high relative velocity compared to the satellite, so the damage should not be so great.

I looked around on the Net and found that a typical BB has a mass of 0.2 gram. A million of them would have a mass of 200 kilograms. You could launch a bunch of them at one time, but the key is to disperse them, perhaps with an explosive charge. That could cause some of the BBs to lose enough velocity to quickly reenter while causing the remainder to spread through space into higher altitudes and different orbital planes. Still, space is big. Every year, we have meteor showers as the Earth passes through the debris trails of comets. I don't know of any serious impacts from these meteor showers that have damaged satellites.

Many pieces of debris don't stay in orbit for very long. They have relatively low mass to surface area ratios, so they quickly lost velocity due to drag and reenter. A BB is roughly spherical (high drag) but also relatively dense, but I think many of the smaller pieces would reenter with a few days to a few weeks depending on the orbital altitude. This has actually been observed on several occassions. Back in the 1960s, there were two or three attempts at a passive communications experiment called the Westford Needles. They tried to disperse thousands of metal dipoles into LEO to use as radio reflectors. The 1961 experiment reportedly failed because the needles didn't disperse. The 1963 experiment failed because the dipoles didn't make good enough radio reflectors. Most of those needles reentered a long time ago, but some of them are still being tracked and reenter from time to time. Also, the debris from Soviet ASAT tests back in the 1970s and 80s would usually reenter quickly, but some pieces would take a lot longer to burn in.

LEO is used for many types of satellites. Deep space is defined as satellites with a period of greater than 225 minutes. Using an orbit calculator, for a circular orbit, The threshold for deep space starts about 5,900 kilometers above the Earth's surface. LEO ranges from the lowest sustainable orbit, roughly 150-200 kilometers above the surface up to that 5,900 KM level. That's a lot of volume to render useless. There are places where the radiation from the Van Allen belts limits satellite operations. The military and national agencies operate those photographic reconnaissance satellites several hundred miles up. Other than that, most of the LEO satellites that come to mind are the sun-synchronous environmental satellites like NPOESS and DMSP, polar orbit communications satellites like Iridium and its competitors, and scientific satellites (including the ISS). Most of the important military satellites are much higher than LEO.