However, you could use a near rhyme or slant rhyme, focusing on the "ill" sound in "ailing":
* Ailing William, feeling chill.
* Ailing William, with a will, to get well.
Alternatively, you could rhyme with William itself, but change the focus:
* Ailing William, sitting still, with a pill.
* Ailing William, with a thrill, for a cure.
Ultimately, the best rhyming word depends on the context and the tone you're aiming for.