The upper bound is not correct. E.g., let $n=101$, $x_1=\dots=x_{100}=1/1000$, $x_{101}=9/10$. Then $$\left(\prod_{i=1}^{n}x_{i}^{x_{i}}\right)\left(\sum_{i=1}^{n}x_{i}^{1/2}\right)^{2}>7>2.$$
More generally, take any $a\in(0,1)$ and any natural $k$, and then let $n=k+1$, $x_1=\dots=x_k=a/k$, $x_{k+1}=1-a$. Then $$\ln\left[\left(\prod_{i=1}^{n}x_{i}^{x_{i}}\right)\left(\sum_{i=1}^{n}x_{i}^{1/2}\right)^{2}\right]=a\ln\frac ak+(1-a)\ln(1-a)+2\ln(\sqrt{ak}+\sqrt{1-a})\to\infty$$ as $k\to\infty$.