5. Let $x, y$ be positive numbers, and $x+y=k$. Try to find the range of $k$ such that the inequality
$$
\left(x+\frac{1}{x}\right)\left(y+\frac{1}{y}\right) \geqslant\left(\frac{k}{2}+\frac{2}{k}\right)^{2}
$$

always holds.
$(0<k \leqslant 2 \sqrt{2+\sqrt{5}})$