Mice entering condo through a large hole in drywall behind gas range
I know this may sound silly but it actually works. Take some cotton balls and saturate them with peppermint oil and stuff them in that hole. Then do as you stated and stuff some steel wool into the hole. I would follow that with some 45 minute dry time sheetrock mud (you might have to use some foam filler first if the hole is really big).

The peppermint oil will keep the rodents away in the short term. Rodents that are familiar with that entry may return to find the steel wool, which they will ingest and eventually die. The sheetrock mud will be a (hopefully) final deterrant for any other rodents to enter at this location.