One way you could show this is realizing that B is diagonizable by similarity transforms. Those (n-k)x(n-k) transform matrices can be extended to k x k by filling off diagonal elements with 0 and diagonal elements with 1 and using the new k x k transform matrix on E won't affect the I_{k}block. Then what you will have is a the product of a unitary matrix transpose with a diagonal matrix, the elements being the eigenvalues of E, and a unitary matrix. The determinant of this is the product of the matrix determinants which is 1 * det(E) * 1.

det(E) is read off as the product of it's diagonal elements which is just 1^{n}* det(B) = det(B)

There are probably easier ways. See this and look for the section on Block matrices