I saw a mouse stuck on a glue trap where I worked several years ago. We employees were horrified that no one could do anything to help it. It was bloody and certainly didn't suffocate painlessly. (Is any kind of suffocation painless?) We don't know how long it was stuck there, but it was still breathing so the janitor took it away and killed it. I know we can't let little critters overrun our homes, but there are certainly more humane ways to get rid of them. I'm now stepping down off of my soapbox.
I hope the kitty is alright, and that your're able to get the glue out of the quilt. We just put too much work into our treasures to have them ruined like that. :(