For years, David Einhorn’s investors didn’t seem to mind his unusual ways—the aloofness toward clients, midday naps, unpopular stock picks, late nights on the town. Until the billionaire hedge-fund manager fell into a slump.

After more than a decade of winning on Wall Street, Mr. Einhorn’s Greenlight Capital Inc. has shrunk to about $5.5 billion in assets under management, his investors estimate, from a reported $12 billion in 2014, and his investments are struggling.

“My...