Inequality involving sum of binomials

Question

Given $x \in \left[0\,;\,1\right]$, is there a closed form solution (or a very good approximation?) for the tightest (i.e. minimal) offset $l$ such that : $$ \sum_{k\in\left[m - l\,;\,m+l\right]} {n \choose k}\,p^k\,(1-p)^{n-k} > x $$ where $$m = (n+1)\,p - 1$$ is the mode, that is the value of $k$ that maximizes the function $k \mapsto {n \choose k}\,p^k\,(1-p)^{n-k}$.