Sarah, all of our pets, our cats, our dogs, our horses, they are all part of our family. If you lost a relative you would not feel silly for grieving, and for many of us, our animals are so much a part of our family that losing them, particularly so suddenly, can cause just a strong and traumatic a response as you experienced last night. I sobbed for three days when my Reynard died. I could not go to work and my then-employers were less than supportive. I didn't care. I had lost the most important being in my life.
You can not blame yourself for Missy's passing. If a vet had been right there with you when the episode commenced, they would not have been able to assist, apart from making her last moments a torment of injections, pushings and proddings. You were with her when she went, and your love travelled with her to the rainbow bridge. She will wait there for you and you will be reunited with her, and you will enjoy the company of many more wonderful, loving animals through the course of your life.
Sleep well little Missy, and Sarah, we all care and are aching with you in your grief.