Unfortunately, Ernest Ka’ai has proven to be a somewhat elusive figure in the annals of ‘ukulele history. If you’re seeking actual arrangements Ka’ai himself performed in concert, you may be out of luck; even recordings of Ka’ai (at least credited or known 'ukulele performances) are few and far between.
John King adapted some arrangements from Ka’ai’s tutor. You can find these arrangements in his famous solos and duets book.
Otherwise, your primary source is likely to be Ka’ai’s tutor. There are 3 versions (1906, 1910, and 1916; the first two are quite similar but the third represents a significant revision). For solo instrumental arrangements, I recommend the 1916. For more information, see thread: https://forum.ukuleleunderground.com/archive/index.php/t-68589.html
'Hawaiian Ukulele. The Early Methods' compiled by Ron Middlebrook for Centerstream Publishers, contains both Ernest Ka'ai's 1916 instruction book and his Hawaiian Methods for Ukulele and Guitar, both containing solo instrumental pieces and both in C-tuning. It is out of print but still available second-hand or in libraries.
'Hawaiian Ukulele. The Early Methods' compiled by Ron Middlebrook for Centerstream Publishers, contains both Ernest Ka'ai's 1916 instruction book and his Hawaiian Methods for Ukulele and Guitar, both containing solo instrumental pieces and both in C-tuning. It is out of print but still available second-hand or in libraries.