Distribution of polymer lengths
Concentrations are a geometric series

First, let's focus on the reactions where one of the reactants is a monomer $\ce{X_1}$:

$$\ce{X_1 + X_n &lt;=&gt; X_{n+1}}$$

and call the equilibrium constant for this reaction $K_{\mathrm{extend}}$.

From this, we can figure out the equilibrium constant for making a polymer from monomers:

$$\ce{n X_1 &lt;=&gt; X_n}$$

$$K_n = (K_{\mathrm{extend}})^{n-1} = \frac{[X_n]}{[X_1]^n}$$

So we can express the concentration of a polymer $X_n$ dependent on the monomer concentration:

$$[X_n] =  (K_{\mathrm{extend}})^{n-1} [X_1]^n = \frac{1}{K_{\mathrm{extend}}}(K_{\mathrm{extend}} \cdot [X_1])^n$$

This is a geometric series (if you take the concentration of polymers as a series, the ratio of neighbors in the series is constant). Because the amount of matter is finite, $K_{\mathrm{extend}}\cdot [X_1] &lt; 1$ for an equilibrium scenario (as opposed to ending up with a single chain containing all the monomers). So the concentration of $[X_1]$ is the highest, and concentrations drop with increase of $n$.

Expression for $[X_1]$

Let $[X]_\mathrm{tot}$ be the total concentration of monomer, no matter whether in a polymer or not.

$$ [X]_\mathrm{tot} = \sum_{j=1}^{\infty} j[X_j]$$  $$ = \frac{1}{K_{\mathrm{extend}}} \sum_{j=1}^{\infty} j (K_{\mathrm{extend}}\cdot [X_1])^j $$

Using $$\sum_{j=1}^{\infty} j \beta^j = \frac{\beta}{(\beta - 1)^2}$$

we get

$$ [X]_\mathrm{tot} = \frac{[X_1]}{(K_{\mathrm{extend}}\cdot [X_1] - 1)^2}$$

which you can solve for the monomer concentration.

Relationship to real polymerization reactions

The polymerization reactions in chemistry and biochemistry that I am aware of are kinetically controlled. There are fibers (such as actin) in biochemistry that are dynamic (i.e. there is polymerization and depolymerization), but the processes are affected by nucleotide binding, complicating matters. I would be interested to hear whether the OP has an example that behaves like this.