what would you like me to do? sounds better to an English ear. I don't think it's correct as it stands.
I've already posted the sentence you suggested (you can see it among the English variants above). So, should I delete the current sentence?
I believe it is a possible wording, although, personally, I use "What would you like me to do?", same as you suggested.
"What would you like me to do?" (which you have) and "What do you want me to do?" both work. "What do you like me to do?" is an unlikely sentence at best.
I believe that there might be situations where this sentence could be said (although you are right, it's not so common to use such a sentence).
What do you like me to do?
Deleted, following comments from various colleagues.
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