1. Define the sets and the problem:
- We start with $K(n,0) = \varnothing$.
- For all nonnegative integers $m$ and $n$, we define $K(n, m+1) = \left\{ k \mid 1 \le k \le n \text{ and } K(k, m) \cap K(n-k, m) = \varnothing \right\}$.

2. **Lemma: For $m \ge n$, we have $K(n, m) = K(n, m+1)$.
- Proof by strong induction on $n$:**
- Base case: If $n = 1$, then $K(1, m) = \{1\}$ for all $m \ge 1$, which is trivially true.
- Inductive step: Assume the lemma holds for all $n \le k$. We need to show it holds for $n = k+1$.
- Fix $m \ge k+1$. For $1 \le i \le k$, by the induction hypothesis, $K(i, m-1) = K(i, m)$.
- Therefore, $K(k+1, m) = K(k+1, m+1)$ because $K(i, m-1) \cap K(k+1-i, m-1) = \varnothing$ implies $K(i, m) \cap K(k+1-i, m) = \varnothing$.
- This completes the induction. $\blacksquare$

3. **Define $S_n = K(n, n)$ for each $n \in \mathbb{N}$:**
- From the lemma, $S_n = \left\{ k \mid 1 \le k \le n \text{ and } S_k \cap S_{n-k} = \varnothing \right\}$.

4. **Conjecture: For positive integers $n$, the set $S_n$ is the set of all integers $m$ such that the unique set of powers of two with sum $m$ is a subset of the unique set of powers of two with sum $n$.
- Proof by strong induction on $n$:**
- Base cases: For $n = 1$ or $n = 2$, this is easily checked.
- Inductive step: Assume the conjecture holds for all $n < k$. We need to show it holds for $n = k$.
- We need to show that for $1 \le \ell \le k$, $S_{\ell} \cap S_{k - \ell} = \varnothing$ if and only if the tasty set of $\ell$ is a subset of the tasty set of $k$.
- If direction: If the tasty set of $\ell$ is a subset of the tasty set of $k$, then $S_{\ell} \cap S_{k - \ell} = \varnothing$.
- Only if direction: We check the contrapositive. If the tasty set of $\ell$ is not a subset of the tasty set of $k$, then the tasty set of $k - \ell$ must not be disjoint with the tasty set of $\ell$. This implies there is a common $1$ in the binary representations of $\ell$ and $k - \ell$, meaning $\ell \in S_k$ and the contrapositive is proven. $\blacksquare$

5. Enumerate the number of positive integers with tasty sets contained in the tasty set of 2004:
- The binary representation of 2004 is $11111010100_2$, which has 7 ones.
- The number of subsets of a set with 7 elements is $2^7 = 128$, but we exclude the empty set.
- Therefore, the number of elements in $S_{2004}$ is $2^7 - 1 = 127$.

The final answer is $\boxed{127}$