This hole is 10" x 9", but as you can see it's crumbling on the bottom, so after removing that it would make it 10" x 14". I was going to insert a couple of ferring strips to attach the new drywall to, but since the damage goes to the floor, should I instead take the drywall off the studs to either side and secure the new piece directly to the studs?
Make it bigger. Particularly, make it bigger so you have half of the stud on each side exposed. Leave the other half for the part of the wall you are not repairing to use.
So yes, secure it to the studs directly.
Funny little secret about drywall repair - a big hole is just as easy to fix as a small hole, or even easier.