Note first that $L$ is naturally an endofunctor on the category of graphs and injective graph-homomorphisms that commutes with filtered colimits.  Let $G$ be any graph such that there is an embedding $i:G\to L(G)$.  This gives rise to an embedding $L(i):L(G)\to L(L(G))$, an embedding $L(L(i)):L(L(G))\to L(L(L(G)))$, and so on.  Let $L^\omega(G)$ be the colimit of $G\to L(G)\to L(L(G))\to \dots$.  Since $L$ commutes with filtered colimits, there are canonical isomorphisms  $$L(L^\omega(G))\cong L(\varinjlim(G\to L(G)\to\dots))\cong \varinjlim( L(G)\to L(L(G))\to\dots)\cong L^\omega(G).$$

Furthermore, if $G$ is connected, so is $L^\omega(G)$, and if $G$ is infinite, $L^\omega(G)$ has the same cardinality as $G$.  Thus to get a connected graph of a given infinite cardinality isomorphic to its line graph, it suffices to give a connected graph $G$ of that cardinality that embeds in $L(G)$.  But this is easy; for instance, complete graphs work.  In fact, any uncountable connected graph works, since a connected uncountable graph of cardinality $\kappa$ must have a vertex of degree $\kappa$, and thus $L(G)$ contains a clique of size $\kappa$.