How many ways are there to fill in the tens digit and hundreds digit of $1\underline{\hphantom{2}}\,\underline{\hphantom{2}}4$ so that it is divisible by $11$?
For integer $abcd$ to be divisible by $11$, then $a-b+c-d$ must be divisible by $11$. The only possibilities for $a-b+c-d$ are $-11$, $0$, and $11$.
Possibility 1: $1-b+c-4=-11 \implies c-b=-8$. This gives us two possible values: $c=0, b=8$, and $c=1, b=9$.
Possibility 2: $1-b+c-4=0 \implies c-b=3$. This gives us $7$ possible values, where $c$ is any integer from $3$ to $9$, and $b=c-3$.
Possibility 3: $1-b+c-4=11 \implies c-b=14$. This is impossible since the digit $c$ cannot be greater than $14$,  Thus there are a total of $2+7=\boxed{9}$ possible values.