To solve this problem, we need to determine the number of ways to choose the sets $S_{i,j}$ such that they satisfy the given conditions. First, consider a fixed set $\{1, 2, \ldots, 2n\}$. We construct nested sets $S_{i,j}$ with $i + j$ elements, ensuring that $S_{i,j} \subseteq S_{k,l}$ whenever $i \leq k$ and $j \leq l$.

### Step-by-step Process:

1. Set Arrangement for Layers:
We deal with $(n+1)^2$ sets $S_{i,j}$ where each set $S_{i,j}$ is required to have $i + j$ elements. The restriction $S_{i,j} \subseteq S_{k,l}$ when $i \leq k$ and $j \leq l$ implies a hierarchical structure:

- Start by choosing a sequence of sets for each $i$ and $j$ such that progressively larger sets cover them due to the increasing number of elements as defined by $k$ and $l$.

2. Choosing Elements:
- We begin by observing that all chosen elements must eventually fit into the largest possible set $S_{n,n}$ which has $2n$ elements (since $n+n = 2n$).
- Each smaller set $S_{i,j}$ requires exact $(i+j)$ elements to be chosen from $S_{n,n}$.

3. Mapping to Combinatorial Problems:
- For any fixed order of choosing $2n$ elements to cover the largest $S_{n,n}$, observe the set hierarchy forms a combinatorial structure where choices within restraints for smaller sets can zigzag across combinations.
- Define binary sequences $b_{i,j}$ representing the inclusion of elements. A binary representation allows flexibility across steps.

4. Counting Possible Arrangements:
- The elements in $S_{i,j}$ must be logically nested within the combinations chosen such that each incrementally larger set respects the nested chain rule.
- The number of ways to permute all the objects and achieve the hierarchy is $(2n)!$. Each permutation describes a unique overall distribution of elements due to the permutation covering the largest set.
- Each level’s inclusion can branch within its inherently defined structure uniquely, resulting in an extra combinatorial factor due to independent binary choices: $2^{n^2}$.

Therefore, the total number of ways to construct such sets $S_{i,j}$ is:

$$(2n)! \times 2^{n^2}$$

Thus, the number of ways to choose $(n+1)^2$ sets $S_{i,j}$ as required is:

$$\boxed{(2n)! \cdot 2^{n^2}}$$