Oh that's so sad, poor mom! It is not your fault though. Put yourself in your mom's place, do you think she feels it's your fault? She probably feels bad that you feel bad.
Accidents happen sometimes that are totally out of our control.
Take comfort in the fact that because you feel bad you have a caring and loving heart and you care about your mom.
I am around older people a lot. And just the mere thought of one of them falling down makes me feel terrible. I would much rather fall instead.
Hope you feel better. It's not your fault. Hey, at least she had you around and she wasn't off somewhere by herself.