Ground ivy aka creeping charlie is a tough aggressive weed.
https://www.extension.umn.edu/yardandgar ... borax.html
suggests borax solution:
"Boron toxicity results in yellow and brown spots around edges of leaves. Stems wilt and eventually the weed dies. Established grass appears to withstand the minute excess of boron needed to kill creeping charlie, though it may show brown discoloration temporarily. If the borax solution is applied in too high a concentration, or repeatedly, it will then be toxic to grass and many other plants."
You would want to be very careful with the borax, it is not only herbicidal, but toxic to honeybees and other beneficial insects. It has to be done in the spring when the plant will uptake it better and it is apparently cumulatively toxic to all plants to some degree, and doesn't have a way to be naturally eliminated from the soil. So I can't say I'm recommending it, but just for info...
For no-harm methods:
It has been suggested that mowing your lawn very short (like scalping it) so that you mow off a lot of the creeping charlie leaves helps. If you just mow high, you cut the grass without touching the creeping charlie, which gives CC an advantage.
Creeping charlie grows best in moist shady spots with low soil fertility, or compacted soil. So anything you can to to improve the drainage, reduce moisture, let in more light
, improve the soil, will help your grass compete better. If you drag a rake along the ground, you can catch a lot of the CC runners and pull it. That doesn't get the roots, but if you do it on a regular basis, it will slow it down.
Otherwise, wait until after the rain when the ground is soft and just keep tracing the runners and pull and dig. It will be a lot of work and you have to be very persistent/ consistent. But if you have a small lawn and you care enough about the CC to keep working at it, you can stay on top of it.