Re-analysis of like (to) (an adjective meaning "likely") as an adverb. (Forms like liked to and would have liked to are from re-analysis as a verb.)
From
Wiktionary
Like-to Sentence Examples
Would you like to go home?
What would it be like to run her hands over Darkyn's lean frame the way she had Gabriel's, to feel his sharp teeth nip the delicate skin of her inner thighs and breasts?
He had indicated enough times that he would like to see her in something a little more feminine and fashionable.