Stationary sets and infinitary logic

Let K⁰_λ be the class of structures (λ, <, A), where A ⊆ λ is disjoint from a club, and let K¹_λ be the class of structures (λ, <, A), where A ⊆ λ contains a club. We prove that if λ = λ ^<κ is regular, then no sentence of L_λ+κ separates K⁰_λ and K¹_λ. On the other hand, we prove that if λ = μ⁺, μ = μ^<μ, and a forcing axiom holds (and א^L₁ = א₁ if μ = א₀), then there is a sentence of L_λλ which separates K⁰_λ and K¹_λ.