The king liked her works so [much/a lot] that he wanted to keep her with him forever.
'much' is correct. But I'm not sure whether 'a lot' is possible or not in that sentence.
If not, I want to know the reason
Let's consider this simpler example. (1) She practiced her swimming so much that she broke the world record. 'so much' here means to the extent that .
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Let's consider this simpler example.
(1) She practiced her swimming so much that she broke the world record.
'so much' here means to the extent that. The phrase is followed by a clause that gives the consequence.
The structure is so much + consequence clause
(2) She practiced her swimming so a lot that she broke the world record