We prove that König's duality theorem for infinite graphs (every graph G has a matching F such that there is a selection of one vertex from each edge in F which forms a cover of G) is inherently of very high complexity in terms of both the methods of proof it requires and the computational complexity of the covers it produces. In particular, we show that there is a recursive bipartite graph such that any cover as required by the theorem is highly non-computable; indeed it must be above (in Turing degree) all the recursive iterations of the Turing jump. This implies that the theorem is proof theoretically at least as strong as the system ATR₀ which is known to be strictly stronger than compactness or König's lemma. Thus the theorem cannot be proven by elementary means plus compactness. Transfinite methods are actually necessary. The actual cover given by the proof considered is seen to have an additional maximality property which makes the assertion of its existence imply a stronger system, Π₁¹-CA₀. We refine this known proof of König's theorem to show that in fact its consequences are equivalent to Π₁¹-CA₀.