A rule of thumb here, not perfect grammar I know but rather a working explanation for intermediate level ESL.
"What..." often works like a single noun:
I know **the answer**.
I know **what you need**.
He always does **the work**.
He always does **what she wants**.
You could replace the "what" clause with "something": "I know something", "He always does something".
In contrast, "That" joins two sentences together, replacing part of one sentence.
This is the house. Jack built it.
This is the house that Jack built.
This is the cat. The cat killed the rat.
This is the cat that killed the rat.
You can't replace the "that" clause with "something". You can't say, unless drunk or gibbering, "This is the cat something."
You can say: "Would you like to do something?", and so you could also say "Would you like to do what others have told you is impossible?"
But you can't say "What would you like to do something?" You'd just say "What would you like to do?" So in this case, a "what" clause doesn't work.