I'm reading an article on second order characterizability. At some point in the article it proves that any model $A$ of cardinality $\kappa$ must be characterizable in $L^2_{\kappa^+\omega}$. I.e. that there is a sentence $\varphi$ in $L^2_{\kappa^+\omega}$ s.t. for any other model $B$, $$B\models \varphi\iff B\cong A.$$
Now, the proof hinges on extending the language $L$ of $A$ by a binary predicate $R$ that well orders $A$ in type $\kappa$. However, the article claims that it's easy to find a sentence $\phi$ in $L_{\kappa^+\omega}$ such that $A\models\phi$ and given any other model $B$ in the language $L\cup\{R\}$ s.t. $R$ well orders $B$ in type $\kappa$, then $$B\models \phi \iff B\cong A.$$
I believe the idea here is to be able to "name" all the elements in the model $A$ (by saying their position in the well order $R$), and then you can write down all the truths of $A$ and put them in one big conjunction and that becomes your sentence $\phi.$ However, what I am not convinced of and need some help proving to myself, is that this is still doable in $L_{\kappa^+\omega}$; in particular, I am worried that the $\kappa^+$ might not be enough. This should be a "simple" cardinality argument around formulae and the like, but I'm not sure how to go about it.
EDIT:
The link to the article: http://arxiv.org/abs/1208.5167
I have tried multiple arguments to no avail.
I believe the number of all possible formulae in $L_{\kappa^+\omega}$ is at least $\kappa^+$, from looking at the cardinality of the set $\{f:\kappa\to |L|\}$, so no way to bound the number of truths in $A$ this way.
Alternatively, I thought about just putting in one big conjunction just the QF or even just the atomic formulae. However, the article makes no mention of the cardinality of the language of $A$, making this potentially also bigger than $\kappa$.
In any case, it seems the cardinality of the language comes into play, am I right, or is that irrelevant? Alternatively, is is possible to argue that if there are "too many" symbols in the language, some of them need to somehow colapse (coincide with one another) given that $A$ has cardinality $\kappa$?
Any thoughts on how I can capture the isomorphism type of $A$ in one single $\kappa$-long sentence $\phi$?
EDIT 2: Think of the structure $$N:=\big(\mathbb{N};(P_i)_{i\in\mathcal{P}(\mathbb{N})}\big),$$ in which each $P_i$ is the obvious one. This is an structure of cardinality $\aleph_0$ and language of cardinality $2^{\aleph_0}$. If I try to trace the proof given in the article, I need to give a well order on $\mathbb N$, so the usual $<$ should work. However, I believe there is no way to get a sentence in $L_{\omega_1\omega}$ that captures the isomorphism type of $N$, even if I have the descriptive power of $<$ available. I need to take a conjunction of $2^{\aleph_0}$ sentences to capture it, right?
Where am I going wrong?