Those are couch grass roots. An absolute pain in the garden, especially when it gets into the root ball of a perennial plant. I would buy some weedkiller containing glyphosate or do as I do, find a farm shop which supplies farmers and buy neat glyphosate. Using protective gloves mix up a measure of the weedkiller and mix with enough wallpaper paste to make a thick gloop. Sort out as much of the grass shoots as you can, tie them in small bundles easing them away from your dierama, put a piece of plastic or cardboard between the foliage of the plant and the weed and work the gloop into the blades of grass. It can take about 3 weeks for the grass to die. You will need to do this continuously until the last of the weed is killed off, no reason to only treat one bundle at a time.
The only other alternative I can suggest is to lift the dierama in the Spring and separate the bulbs from the grass. You will need to remove every blade of grass and every piece of root as couch grass will regrow from any tiny piece of root. I personally would wash the bulbs to remove all soil to avoid carrying any grass seeds and plant in a new location.
I have a similar problem with another grass, no idea what it is called. It grows from bulbils , looks vaguely like a miniature bamboo and it smothers anything within a couple of feet from where it is growing. It makes large clumps, achieving around 4ft in height and comes in from the uncultivated fields surrounding my garden. The juvenile bulbils are miniscule, like pills of soil so is really difficult to eradicate. Similar to celandine, another pet hate of mine.