Show that for any measurement operator $M_m$ there exists unitary $U_m$ such that $M_m=U_msqrt{E_m}$ with $E_m$ POVM

Exercise 2.63 of Nielsen & Chuang asks one to show that if a measurement is described by measurement operators $M_m$, there exists unitary $U_m$ such that $M_m = U_m sqrt{E_m}$ where $E_m$ are the POVM associated to the measurement (that is, $E_m = M^{dagger}_m M_m$).

I can see that, if $sqrt{E_m}$ is invertible, then $U_m = M_m sqrt{E_m}^{-1}$ is unitary; indeed, we have (dropping the needless subscript for simplicity) $U^{dagger} U = (sqrt{E}^{-1})^dagger M^dagger M sqrt{E}^{-1} = (sqrt{E}^{-1})^dagger E sqrt{E}^{-1} = (sqrt{E}^{-1})^dagger sqrt{E} = (sqrt{E}^{-1})^dagger sqrt{E}^dagger = (sqrt{E} sqrt{E}^{-1})^dagger = I^dagger = I$
where I used that $sqrt{E}$ is Hermitian (since it is positive).

But what if it’s not invertible? Perhaps some continuity argument would work?