While Hawaiian Paradise (1959) is a classic slice of vintage tiki sounds its background is a bit of a mystery. The only documented release of this LP is on a British label, Fidelio, but it appears to be American in origin due to the presence of arranger and conductor Jim Timmens and vocalist Anita Darian.
Timmens made several jazzy LPs for Warner Brothers in the late 1950s, including the highly regarded Gilbert and Sullivan Revisited. He also worked on a bunch of children's recordings, some in the company of Darian, an operatic style soprano who graces a few of the tracks on Hawaiian Paradise. She first gained notice with the Sauter-Finigan Orchestra in the mid-'50s.
As for Bill Jaffee and His Islanders, this is likely a moniker made up for this studio recording, as there are no other releases under Jaffee's name. Here's one of the instrumental tacks from the LP.
House of Jade
Hawaiian Paradise is not available in digital form, but used copies of the LP have been known to turn up on eBay, Amazon and elsewhere at reasonable, and sometimes downright cheap, prices.