His success in the United States has made Mr. Eklund, a member of a prominent Swedish family, with ties to prime ministers, the royal family, even the legendary director Ingmar Bergman, somewhat of a minor celebrity back in his native Stockholm, and the subject of glowing profiles in the Swedish press. But it’s more than just his ability to close million-dollar deals that has brought him that fame. There’s also that other career of his: that of a one-time actor who starred in a half-dozen adult...