Not possible
TL;DR
In order to ensure that an ICBM had only a 5% chance of surviving for 20 minutes in space, you need to launch at least 5 orders of magnitude more mass into space than has been launched since Sputnik. This is not economically feasible, so Kessler syndrome cannot protect you from ICBMs.
A note on ICBMs
First we have to define what an ICBM is going to do. It travels a ballistic course, up to 1200 km altitude, and it can be launched from any latitude. This part is important; a sub-launched missile from the North Pacific can hit Moscow while never travelling below the 50th parallel north.
What is the collision frequency of orbital objects?
From the original paper, Kessler and Cour-Palais, 1976, the collision rate for a single object in orbit is
$$ \frac{dI}{dt} = S\bar{V}_sA_c,$$
where $I$ is the cumulative number of impacts, $S$ is the spatial density of particles in orbit in km$^{-1}$, $\bar{V}_s = 7 \text{km/s}$ is the relative impact velocity (calculated in the paper), and $A_c$ is the cross-sectional are of the object in question.
Generously counting the third-stage booster, a modern Trident II missile has a 7 foot diameter and is about 12 feet long. Lets use a 7x12 foot surface as cross sectional area, which works out to $8\times10^{-6}$ km$^2$ to match units. This gives us an equation:
$$ \frac{dI}{dt} = 5.6\times10^{-5} S$$ where we will solve for $S$ in units of particles per cubic kilometer and $dI/dt$ in collisions per second.
What collision rate leads to a 5% survival rate for an ICBM?
A Poisson distribution gives the number of discrete events in a certain time period, based on a rate function, the probability that the event will happen. The probability mass function for the Poisson distribution is $$P = \frac{\lambda^k e^{-\lambda}}{k!}.$$
$P$ is the probability that an event will happen, $\lambda$ is the rate of an event happening, which is what we will solve for, and $k$ is the number of events. We want to solve for $P = 0.05$ and $k=0$; that is, there is a 5% chance that there will be zero collisions. Plugging these in, we solve for $\lambda=3.00$.
For our Trident missile, let us assume 20 minutes (1200 s) are spent in the orbital regions of space. Therefore $\lambda$ is in units of events per 1200 seconds. Converting to units of events per second, we get $\lambda = 0.0025$. This number is equivalent to the $dI/dt$ that we want.
How much space do we need to fill with particles?
Mentioned in the introductory note is the point that ICBMs can get into all sorts of unusual parts of space, because they are sub-orbital. They can fly right across the north pole. Orbits that take our anti-ICBM particles into high latitudes will also inevitably cross the equator, where there will also be anti-ICBM particles that are orbiting at low latitude. Thus, there will be a distribution of satellite orbits that peaks at the equator and reduces in density near the poles.
I'm going to ignore that; instead I'm just going to say we need a constant density debris field over the Earth. We'll see why in a minute. In order to have a good chance of stopping an ICBM, we need density $S$ of particles in all parts of space that an ICBM might reach over its 20 minute stay. This is a shell from 300km to 1200km from the surface of the Earth, or at a radius of 6670 to 7570 km.
$$ V = \frac{4}{3}\pi(7570)^3 - \frac{4}{3}\pi(6670)^3 = 5.7\times10^{11} \text{ km}^3.$$
How much mass of particles do we need?
Plugging $dI/dt$ into our collision rate equation, we get S = 44.6 objects per cubic kilometer of space. Multiply this by the space we need to fill and we need 25 trillion particles. As you can see, increasing the particle density near the equator is not really necessary; this is plenty of ball bearings to launch.
Generously assuming a 100g particle can kill a ballistic missile (doubtful!), we need to put 2.5 trillion kg of particulate mass into orbit to allow only a 5% chance of an ICBM to get through. The total amount of spacecraft launched since the dawn of the space age is roughly 13 million kg.
Conclusion
Kessler syndrome is dangerous on the timescale of weeks and months. Once there are enough particles, your spacecraft won't last out a year, and then it is hard to justify launching something into orbit. But the particle density required to stop an ICBM, with a fast, suborbital path through space is immense. Especially with mobile ICBM platforms (i.e. submarines) nearly any trajectory for a ballistic missile is possible, so you would have to cover everything to be safe.
Even dropping the requirement to a 50% kill-rate and redoing all the calculations only drops the mass requirement by a factor of about four. You still need roughly five orders of magnitude more mass in space than has been put there in the past 50+ years to form an effective defense. Kessler syndrome cannot effectively guard against ballistic missiles.
