Expected norm-squared of one random vector projected onto others, all iid

Question

For $k\leq n$, $x_1,\dots,x_k \in \mathbb{C}^n$ are independent identically distributed random vectors almost surely unit norm and with span dimension $k$. Call $X=[x_2,\dots,x_k]$. I am studying $\mathbb{E}[\| X(X^\dagger X)^{-1} X^\dagger x_1 \|^2]$ but the inverse is making this challenging. Is there a series expansion in terms of $X$, or another approach to this quantity?