Your solution is correct. The official answer is wrong.
The question should ask for the maximum distance, not minimum. The closer you are to the mirror the more of your image you can see at any angle of tilt. That error might alert you to the possibility that the official answer is also wrong.
If you find that you don't have the "right" answer, and you cannot spot your mistake, the next thing to do is check the reasonableness of both answers. What does each answer predict when you input values which you know the answer to?
The obvious trial value here is $\theta=0$. Then you can always see your feet, at any distance from the mirror. Your answer predicts $d=\infty$, which is what you expect. The official formula predicts $d=\frac12 h$, which is not what you expect. A sketch shows that you can still see your feet closer or further than $d=\frac12 h$.