Marigold,
You didn't let this little sweetheart down, you helped her. I don't know her age, but it sounds like she was pretty small and maybe still needing her mommy. Had you not picked her up, held her with love, and went to every place you could think of to find a vet for help, how much longer might she have suffered in pain and fear? You did what was best for her but broke your heart in the process.
She doesn't hurt anymore and just maybe she found her mommy waiting for her at the Bridge.
Because of you, this special little being knew that humans give comfort and love. Bless you for that and bless you for caring enough to help