I leave stems and leaves to die down in their own good time. Your daylily's got buds and will bloom again for you pretty soon. Daylilies are very easy to grow. They need a bit of fertilizer in spring. I usually fertilize when I remove the dead leaves and stems. Then they just sit there and do their stuff until they die down in autumn.
I also just let the stems die off on their own. Be sure to let us see how pretty it is when it blooms again.
I, too, just let the flower stalks and leaves die back on their own. Mine are in bud at the moment so, hopefully, we'll get some good weather to help them burst into bloom.
If there are no buds, instead of letting the foilage slowly wither and turn brown, get out the clippers and cut the stems and foliage down to about an inch. Daylillies have a very good response to liquid feeding. In about two weeks, the foliage will regenerate and look very nice and green. Much better than having an ugly looking brown mess.
