A few answers with references:

The answer to the original question is "yes": There are simple groups of any infinite cardinality. For example, the set of all permutations of $\kappa$ with finite support has normal subgroup $A_\kappa$ of index 2, the even permutations; $A_\kappa$ has cardinality $\kappa$ and is simple.

More examples of simple groups of any infinite cardinality can be found in Lang's book "Algebra", chapter XIII, sections 8 and 9.

For the special case of $\kappa = 2^{\aleph_0}$, $\lambda=\aleph_1$, wikipedia mentions Ch 11.3 in Scott's 1987 book on Group Theory and Ch. 8.1 in Dixon-Mortimer's 1996 book on permutation groups as references for the following theorem:

• The symmetric group on a countable set has only 2 nontrivial normal subgroups, both of them countable: even permutations, and permutations with finite support.