Smithy,
I would suspect you have Ruellia squarrosa or one of its close kin. Very invasive! I once saw a garden pond where it was growing around the edge and the roots were thriving in the water in a mass, tangled netting. This plant also likes water.
Try a weed killing product such as Round-up. Because you are probably trying to kill it off in the area of other garden plants I would suggest that you 'paint' the tops of the Ruellia using an old paint brush. In more open sections of the garden use a small paint roller for faster coverage. This method gives you complete control over where the weed killer goes and should not harm desired plants. Yes, the process takes time (and sore kness) but is the safest. Withhold water after treatment for as long as possible. Try a test area and see if it works. Usually with invasives such as this you will be spot treating for several years to get a clean kill.

There are a lot of Ruellia species in the Americas. In Miami, Florida, I found that R. squarrosa was manageable. Our local form of the native R. caroliniana looked similar but spread quicker vegetatively and threw seeds ten feet or more from the parent. Roots left in the ground can sprout new shoots. Nandina is right that glyphosphate or a lot of manual work will be needed.