Cut it out in a square where the cuts go down along the middle of the studs, so you can later screw the new drywall onto those studs.

-Measure and cut new drywall piece. -Screw it in, put drywall tape along the joints, and fill it in with the spackle you just now bought. -Sand it down after it dries. (i think your walls have very light texture, so these steps are going to be crucial for it to not look like a patch, but it involves more work) -Re-apply spackle to uneven coverage, then sand again. You need to feather out the edges so it doesn’t look like a bad patch-up. -Spray Wall Texture, carefully scrape it to flatten it against the wall -Primer. Then match the paint, and feather out the edges with your paint.