I personally, would not use any phrases to a child that link sleep with death. Children are much more logical than we give them credit for, so I would explain that part of being a responsible loving pet owner is to feed them properly, look after them when they are ill, take them to the vet, etc. And, that making this final decision is the last loving thing we can do for them because we don't want them to suffer anymore.
Yes, I'm sure she will be upset, but she will also understand why. She will grieve but that is part of the process of losing any pet, or anyone we love. It is upsetting for us as loving adults to see children upset, especially about the loss of a beloved pet when we may be grieving ourselves, but again it is part of our job as adults to support a child through this very natural reaction, helping a child to process loss, not to try and minimise it or avoid it.