Ground ivy is very hard to kill but this should do it. Only do a max of two applications per year (spring-fall). Go to a local feed mill and ask for the broadleaf herbicide Trimec Classic or Triplet. Do not get Trimec Plus, it has 1/2 the salt of dicamba in it that Trimec Classic has.
I apply 3 oz per 2 gal water in a pressurized 2 gal hand sprayer to 1000 sq ft. Triplet is about $37/gal and Trimec is $45/gal