The compactness number for $\mathcal L_{\kappa,\kappa}$ is equal to the least $(\kappa,\infty)$-strongly compact cardinal. A cardinal is $(\kappa,\infty)$-strongly compact if for every set $X$, there is a $j : V\to M$ such that $\text{crit}(j)\geq \kappa$, and $j[X]$ can be covered by and element of $M$ of $M$-cardinality less than $j(\delta)$. I sketch a proof at the end because I don't know the reference.

But first: it follows easily that your hypothesis is equivalent to the existence of a proper class of almost strongly compact cardinals, which are (resp. should be) defined to be cardinals $\kappa$ such that for all $\gamma < \kappa$ every $\kappa$-complete filter can be extended to a $\gamma$-complete (resp. $\gamma^+$-complete) ultrafilter. Whether this is equivalent to the existence of a proper class of strongly compact cardinals is an open question. 
The true consistency strength is probably a proper class of supercompacts: all three of these hypotheses are equivalent under the Ultrapower Axiom. There is some evidence that the equivalence between a proper class of almost strong compacts and a proper class of strong compacts is a theorem of ZFC: the first almost strongly compact cardinal above an ordinal $\gamma$ is either strongly compact or else has countable cofinality (although the truth is I needed a little SCH to handle the least one). This is in [Some combinatorial properties of Ultimate $L$ and $V$][1].

Now the proof. In one direction, you show that $\mathcal L_{\kappa,\kappa}$ is $\delta$-compact for any $\kappa$-strongly compact $\delta$. Fix a $\delta$-consistent theory $T$ in the signature $\tau$. Cover $j[T]$ by a theory $S\subseteq j(T)$ in $M$ of $M$-cardinality less than $j(\delta)$. You get a model $\mathfrak A$ of $S$ in $M$ by $j(\delta)$-consistency of $j(T)$. Take the reduct of $\mathfrak A$ to $j[\tau]$. This is essentially a model of $T$: more precisely, $j : T \to j[T]$ is an isomorphism of $\mathcal L_{\kappa,\kappa}$-theories because $\text{crit}(j)\geq \kappa$. Conversely, if $\delta$ is the compactness number of $\mathcal L_{\kappa,\kappa}$, then for any $\gamma$, you can build a $\delta$-consistent theory whose models are $\kappa$-complete fine ultrafilters on $P_\delta(X)$. The theory has names for all subsets of $X$. Compactness yields such an ultrafilter $\mathcal U$, and the associated ultrapower embedding $j : V\to M$ has critical point at least $\kappa$ and closure under $\kappa$-sequences by $\kappa$-completeness. Finally $\text{id}_\mathcal U$ is a cover of $j[X]$ by fineness, and $\text{id}_\mathcal U$ has $M$-cardinality less than $j(\delta)$ since it is an element of $j(P_\delta(X))$ by the definition of $M$-membership.


  [1]: https://arxiv.org/abs/2007.04812