There is a subtle point about this phrase.
The correct way to say "I don't know what to choose" is 어떤 것을 고를 지 모르겠어요.
"... to (choose)" usually translates to "-ㄹ지".
-ㄴ지 corresponds to present and present progressive tense conjugation, distinct from -ㄹ지.
It is a small but important difference that changes the meaning.
See these examples.
I don't know what to do. 나는 어떻게 (무엇을) 해야 할지 모르겠다. (-ㄹ지 = to...)
I don't know what I'm doing. 나는 무엇을 하고 있는지 모르겠다. (-ㄴ지 = present progressive)
Is there anything he has to do? 그가 해야 할 일이 있어요?
Is there anything he is doing / he does? 그가 하는 일이 있어요?