Final answer:
The external urethral sphincter, under voluntary control, is the muscle responsible for urinary continence that the elderly client can no longer control, leading to incontinence.
Step-by-step explanation:
The muscle that an 89-year-old client with acute renal failure and incontinence of urine is no longer able to control voluntarily, leading to incontinence, is the external urethral sphincter. This sphincter is composed of striated muscle and is under conscious control via the somatic nervous system. The inability to control this sphincter results in the loss of urinary continence. Voluntary control of urination, or micturition, relies on preventing the relaxation of the external urethral sphincter, which, when relaxed or unable to be controlled, leads to incontinence.