This study considers the problem of state-derivative feedback controller design via eigenvalue assignment for LTI systems of linear Delay Differential Equations (DDEs) with a single delay. Unlike simple LTI systems, the systems described by DDEs have an infinite eigenspectrum and it is not feasible to assign all closed-loop eigenvalues. The paper proposes a method to assign a critical subset of them using an approach of the matrix Lambert W function. The solution has an analytical form expressed in terms of the parameters of the DDE. With the proposed method, one can extend a conventional eigenvalue assignment method for a feedback controller to a delayed LTI system. A scheme including filtered state-derivative feedback is proposed to overcome the destabilizing effect of feedback delays. Proofs of the proposed method and numerical examples are presented.