One method that seems to work is to dig up the entire lawn to a depth of 3 inches and remove every piece of root. Apparently that stuff forms a dense web or roots underground. Wait? That wasn't what you wanted to hear was it? Seriously, that method has been reported here and elsewhere over the years as being a last resort after all else has failed. I don't know what speed zone is but if it is not glyphosphate, then you might try Round Up. You are going to have to dedicate your existence to getting rid of it. Forget about trying to manage grass at the same time. This is a war to get your soil back. Once you have it back then you can worry about the grass.

Killing Ground Ivy is relatively easy, if you do it at the right time. The "right time" is when it is actively growing. Right now it's in summertime survival mode. It will resume actively growing and be pretty easy to kill with 2-3 applications of Triclopyr, which is the active ingredient in Ortho Clover, Chickweed and Oxalis Killer (nicknamed "CCO"). Only the 8% stuff will work - weaker solutions are a waste of time. The applications should be about 14 days apart.

Try some 2,4-d, its safe on centipede but use only if the Ivy is activly growing. You might have to wait till spring. http://aggie-horticulture.tamu.edu/Plantanswers/turf/publications/centipede.html Here is a link that might be useful: Centipede

I'm a practitioner of "Integrated Pest Management" - whatever works. I would get rid of the Creeping Charlie by the least invasive method possible, then work on a routine that will keep it out. It likes moist areas ... make sure you have good drainage and aren't overwatering. Use a moisture meter and adjust your watering based on need instead of calendar. Make sure you aren't mowing too short, and fertilize as needed to get a good dense turf. Use the right kind of grass for the conditions. If it's really well-established, and it sounds like it is, use whatever herbicide that is most likely to kill it, exactly as specified on the label. Then carefully remove the corpses and reseed the bare areas. In the future, work on keeping the lawn conditions so they aren't as hospitable to the weed. Check for sprouts weekly and pull them out.