关于$x$的方程$2x-mn+1=0$的解是$x=\frac{mn-1}{2}$，

关于$x$的方程$x-n-1=0$的解是$x=n+1$，

$\because$关于$x$的方程$2x-mn+1=0$与方程$x-n-1=0$的解之和为$5$，

$\therefore \frac{mn-1}{2}+n+1=5$，

$\therefore \left(m+2\right)n=9$，

$\because m$、$n$皆为非负整数，

$\therefore \left\{\begin{array}{l}{m+2=1}\\{n=9}\end{array}\right.$或$\left\{\begin{array}{l}{m+2=3}\\{n=3}\end{array}\right.$或$\left\{\begin{array}{l}{m+2=9}\\{n=1}\end{array}\right.$，

解得$\left\{\begin{array}{l}{m=-1}\\{n=9}\end{array}\right.$（不合题意，舍去）或$\left\{\begin{array}{l}{m=1}\\{n=3}\end{array}\right.$或$\left\{\begin{array}{l}{m=7}\\{n=1}\end{array}\right.$，

$\therefore$符合条件的$m$之和为$1+7=8$，

故答案为：$8$.

关于$x$的方程$2x-mn+1=0$的解是$x=\frac{mn-1}{2}$，

关于$x$的方程$x-n-1=0$的解是$x=n+1$，

$\because$关于$x$的方程$2x-mn+1=0$与方程$x-n-1=0$的解之和为$5$，

$\therefore \frac{mn-1}{2}+n+1=5$，

$\therefore \left(m+2\right)n=9$，

$\because m$、$n$皆为非负整数，

$\therefore \left\{\begin{array}{l}{m+2=1}\\{n=9}\end{array}\right.$或$\left\{\begin{array}{l}{m+2=3}\\{n=3}\end{array}\right.$或$\left\{\begin{array}{l}{m+2=9}\\{n=1}\end{array}\right.$，

解得$\left\{\begin{array}{l}{m=-1}\\{n=9}\end{array}\right.$（不合题意，舍去）或$\left\{\begin{array}{l}{m=1}\\{n=3}\end{array}\right.$或$\left\{\begin{array}{l}{m=7}\\{n=1}\end{array}\right.$，

$\therefore$符合条件的$m$之和为$1+7=8$，

故答案为：$8$.