I thought that すれば only stood for the conditional form, how come does it bear the same meaning as すればいい ?
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In speech, this is a perfectly valid question, and semantically it means exactly the same thing as "私は何をすればいい((の)ですか)?" The last part is omitted because it's obvious.
To make the first sentence sound valid and natural, all you have to do is use the rising intonation at the end of the sentence to indicate it's a question. Maybe this could sound slightly blunt, cold or business-like if said in a non-urgent situation. But this would largely depend on how the character usually speaks.