High prey drive dogs and cat cohabitation

For those of you who have successfully managed a high prey drive dog and trained them to live peacefully with cats… how?

I have an amazing 5-month old puppy… but she has a high prey drive and gets really worked up about the cats (we have 4)… I adopted her less than 3 weeks ago and have been trying to only allow her into the room with them when she is leashed, tired, and reinforce positive interactions and calm behavior but if the cats sneak into our space when I’m not prepared she will chase them.

Help!