Losing a pet is always hard, especially one you have had for so long. We have always had funerals for the pets we have lost, and it is a good idea for you to do so, so that your daughter can understand what has happened. It will mean a lot to her if she has a chance to say goodbye to your cat, and it will help her also gain a basic understanding that death is a natural event, not something to be hidden or feared. On the other hand, if you don't let her know what has happened to the cat, she will be more upset, because it would have just disappeared. However, when you are explaining it to her, don't use the words "put to sleep" because that may cause her to be afraid to go to sleep. You should just tell her the cat died because it was very old and sick, and then explain a little bit about death, whatever you feel like she can understand. If you have Christian beliefs, you can tell her that the cat went to Heaven. I would not let her go to the vet's office with you to have your kitty put down, because that could cause her to be afraid to go to the vet, or even to the doctor because after all, the vet is an animal doctor.
I'm sorry for your loss, and good luck with whatever you decide to do in regards to explaining to your daughter what is going on.