The NFA for the given regular expression 1*(0+10)*1* is given below
To find the shortest string which is not accepted by the above NFA we have to find those states which do not have transition for some symbol.
Note that after reaching q3 there is no transition for 1. So,if we have the string 011 then we reach the state q2. And from q2 there is no transition for 0. So, the string 0110 is not accepted.
The shortest string which is not in the language is of length 4.
It is the string 0110.