We first need $\frac{p(p+1)}{2} \le 5^7$, or $p(p+1) \le 156,250$. It follows that $(p+0.5)^2 \le 156,250.25 < 160,000 = 400^2$. Hence, $p < 400$. We know that $p$ divides $5^7$ or $v_2{(p)} = v_2{(5^7)} + 1$. Hence, the only other possibility is that $p$ is an even number such that $p$ divides $2 \times 5^7$. The largest valid odd positive integer less than $400$ is $125$. The largest valid even positive integer less than $400$ is $250$. One can check that in both cases, $(p+0.5)^2$ is not even close to $156,250.25$. Hence, the desired answer is $\boxed{250}$.