The fact that you're worried about being a bad mom should be a clue that you aren't one.
This is a tough age to parent any kid but a gifted, hungry brain that can't speak fluently yet makes it even harder. Mine are 5 and 10 and I still haven't figured out how to like playing with them (I get bored), attending to their constant chatter, and doing housework. Mine are lucky to have an aunt who does pretend play with them and I'm always reminding them that this is her thing, not mine.
For my first, we had a playgroup for a few kids where 2 moms stayed to watch them, and the rest got an hour off. It was really a revelation. I could prep myself to be 'on' for an hour when it was my week - plan painting or whatever - and then on my hour off, sometimes I would just sit in my car and cry, or take a book to a cafe. It kept me sane. Having the second mom around means that one can handle a crisis, while the other watches the kids who are doing fine. Added bonus: with fewer moms, the kids began to interact with each other as peers much more. Their social skills (OK, we are talking 1-3 year olds here) really soared.
I do much better if I can plan ahead, so activities like these worked for me. I spent an hour or two one evening making up activity kits, then had something to pull out to keep her busy and let me catch my breath:
http://www.redshift.com/~bonajo/preschool_activities.htm
It also helped me to think of self-entertainment or independent play as a skill to be taught. I was not being selfish or a bad mom, I was *teaching* her something useful! It takes a while, so set tiny goals at first. Give her washable markers and a paper, and ask her to make you a surprise drawing. 30 seconds later, ooh and aah and ask her to add some more details or make you another... Eventually you work up to a more comfortable amount of time.
And keep reminding yourself that it gets better.
