Proof for values of d with d:= ||L|| - N(L) with $d \in \mathbb{Z}$ and N(L) Nerode Index
In order to make the question nontrivial, you have to assume that $L$ is finite. You haven't specified what the alphabet is, so I will assume that the alphabet is arbitrary.
First, let us show that $N(L) \leq \|L\|+2$. We can construct a DFA for $L$ whose states consist of all prefixes of words in $L$, together with a failure state. This DFA contains an initial state (corresponding to the empty prefix), a failure state, and one state for every non-empty prefix. In particular,
$$
N(L) \leq 2 + \sum_{w \in L} |w| = 2 + \|L\|.
$$
In other words, $d \geq -2$.
Second, let us consider the language $L_\Sigma = \{\sigma : \sigma \in \Sigma\}$ over an alphabet $\Sigma$. The minimal DFA for this language contains three states, and so the value of $d$ for $L_\Sigma$ is
$$
\|L_\Sigma\| - N(L_\Sigma) = |\Sigma| - 3.
$$
As $\Sigma$ ranges over all possible alphabets, we obtain all values which are at least $-2$.