This is a sequel to papers by the last two authors making the Riemann-Hilbert correspondence and isomonodromy explicit. For the degenerate fifth Painlev\'e equation, the moduli spaces for connections and for monodromy are explicitly computed. It is proven that the extended Riemann-Hilbert morphism is an isomorphism. As a consequence these equations have the Painlev\'e property and the Okamoto-Painlev\'e space is identified with a moduli space of connections. Using MAPLE computations, one obtains formulas for the degenerate fifth Painlev\'e equation, for the B\"acklund transformations.