We've had catnip for 30 years and it has endured our cats and
neighbors cats. Nothing kills it. We wack it back severely every
fall and it regenerates every Spring and endures another growing
season's worth of abuse from the many cats who visit our herb
garden. Ours has never grown very tall, but it appears that it
would spread out to horizontally to infinity if we allowed it.

I'd suggest one of two things:
1) Let the cats enjoy it and let the plant adapt.
2) Toss some moth balls on the ground around the catnip
if you are into catnip but not into cats.

In either case, you should be relatively pleased. Catnip is a
very hardy plant.

Good luck,
Gideon