Finding Delta Algebraically for a Given Epsilon?

You know that if $-3 < x-5 < 3 \implies -3 < x-5 < 5$. This means you can take $\delta = 3$.

Assuming your algebra is correct (I didn't check it carefully), you have shown that $|\sqrt{x-1}-2|<1$ if and only if $-3<x-5<5$. Now you want to find $\delta>0$ so that if $|x-5|<\delta$ then $-3<x-5<5$. But $|x-5|<\delta$ is the same as $-\delta<x-5<\delta$. So what is a suitable value of $\delta$?