\(\sqrt{5}=\left|z-1+2i\right|=\left|z+1+i-\left(2-i\right)\right|\ge\left|\left|z+1+i\right|-\left|2-i\right|\right|\)
\(\Rightarrow\left|\left|z+1+i\right|-\sqrt{5}\right|\le\sqrt{5}\)
\(\Rightarrow\left|z+1+i\right|-\sqrt{5}\le\sqrt{5}\)
\(\Rightarrow\left|z+1+i\right|\le2\sqrt{5}\)