Reduction from Exact Cover to Fixed Exact Cover
Add $n$ empty sets, and choose $K = n + 1$. Every solution for the original problem is (without loss of generality) of size between $1$ to $n$, so using the empty sets you can complete it to a solution of size exactly $K$.

If you're determined to have your collection duplicate-free, add $n$ new elements $y_1,\ldots,y_n$, and the following sets:
$$
\{y_1\},\{y_2\},\ldots,\{y_n\},\{y_1,y_2\},\{y_1,y_2,y_3\},\ldots,\{y_1,\ldots,y_n\}.
$$
There are exact covers of the new elements of any size from $1$ to $n$.