How can I simplify $\sqrt{\frac{5+\sqrt{5}}{2}}$? I've tried to see the root as
$\sqrt{\frac{5+\sqrt{5}}{2}} = \sqrt{a}+\sqrt{b},$
but this method doesn't give me something good.

It can't give you anything good.
Say you have $\sqrt{a+\sqrt{b}}=\sqrt{x}+\sqrt{y}$, all numbers rational and $b$ not a rational square.  Then
$a+\sqrt{b}=(\sqrt{x}+\sqrt{y})^2=(x+y)+2\sqrt{xy}$
The uniqueness of quadratic surds forces $a=x+y,b=4xy$.  Then
$a^2-b=(x+y)^2-4xy=(x-y)^2,\therefore\text{ a rational square.}$
But here, $a=5/2$ and $b=5/4$, giving $a^2-b=5$.  So we're having a bad day.
