Cordelia's name and plot is picked from Spenser's Fairy Queen. Gloster's sons and blindness and the cliff suicide may be borrowed from the king of Paphlagonia in Sir Philip Sidney's Arcadia. All the demons mentioned are topical references to "A Declaration of egregious Popish Impostures" by Samuel Harsnett (Shakespeare implying his characters being catholics is where the theory that he himself has been a catholic is taking root). There were some previous plays about King Lear and the king and his daughters also get their mention in ballads and fairy tales which, however, were collected after Shakespeare was long dead. We do know some ballads come from Shakespeare: there's a ballad of Titus Andronicus spreading evidently out of London.

From references in other plays we know he read Holinshed's chronicles and Holinshed does mention king Lear (spelled Leir) briefly as the founder of Leicester.

Source:

Anders, Henry R. D. "Shakespeare's Books; a Dissertation on Shakespeare's Reading and the Immediate Sources of His Works." New York: AMS, 1965.

Kind Lear's story is found in the Historia Regum Britanniae (history of the kings of britian) written by Geoffrey of Monmouth. That book along with the Holinshed Chronicles were the two history books Shakespeare used for his plays. The play keeps a pretty similar account to Geoffrey's. So since Shakespeare had a public education its quite likely that his audiences also knew about king Lear. So the audiences weren't aware that Lear was likely mythic because Lear likely wasn't mythic. The only times that it is believed that Geoffrey went into fiction with his book was when he talked about Merlin and king Arthur. Otherwise historians and scholars agree with Geoffrey's account.