Hi Hywel!

Liriope muscari, or Lilyturf, is usually tough and reliable, but I am beginning to suspect that it is comparatively short-lived, hence the reason why it is not more widely grown. I had two, both of which have simply vanished.
As yours has flowered, but is still alive, might I suggest the following?
Dig it out, split it, possibly into four as it seems quite large, give it a light trim removing the dead flower stems, and replant the pieces in a fresh site. Also, check the roots for cockchafer grubs and leatherjackets. It is a grass, after all, and leatherjackets love grass! Hope this helps.

I would ruffle through the leaves looking for the little tufts of this summer's new growth at the base, and cut off everything else. It would probably also like a moderate feed of washed seaweed, too.

Thank you both ....