Age 6 is really too young for most classes. Very few kids have the attention span or the desire to do something every week. A lot of parents are spending precious dollars on programs, and then here on Mamapedia we see all the questions about what to do when kids don't want to go or who cry when they are there. I think there are a few kids who dearly love these programs, but many more who just aren't ready. We tend to over schedule our kids and it's a huge debate in the child psychology and early childhood education fields.
When they come home from kindergarten, they don't want more structure in a heavy-handed way. But narrowing it down to 2 choices can help young kids, especially those who are fearful or who have trouble making decisions. It may help to organize her toys into sections or specific bins in this corner of the family room or that corner of her room. If you are limiting her screen time (TV or computer), that's a good thing - you can give her a specific time if you know you will be doing things with the baby from 6 to 6:30 every day (or whatever), or you can limit her to 1 DVD a day, whatever is appropriate.
Otherwise, you might have some outdoor time every day (weather permitting) - put the baby in a stroller with a big blanket and let her watch you two outside, or go out during the baby's nap while Dad is sleeping but available if the baby wakes up. If she doesn't like soccer, it may be the organized team thing but maybe she'd like to kick a ball around. Or you can play catch (try the mitts with velcro and a special ball), ride bikes, throw a frisbee (sometimes hard for young kids), go on a nature walk, build an ant farm, collect pine cones for peanut butter & bird seed feeders, make a snow man, and much more. Get some driveway chalk and make a hopscotch game or draw "roads" and "crosswalks" and traffic signs she can then navigate with a bike or roller skates or a doll in a stroller. It reinforces traffic safety and even spelling, while being fun and creative.
The baby's nap time can be your daughter's special time with you. If you are indoors and the baby is napping, then that can be quiet time for your and your older: reading books, doing puzzles, doing art, etc. She can, and must learn to, do things on her own without you. I'm guessing that, in her classroom, the teacher has stations for different things, and the kids are not supervised one-on-one all the time. So you can try to duplicate that at home - there are areas for doing certain things, and you can even direct her to one area or her choice of two, but once there, she plays on her own.
Do you have toys that require thought and creativity? I'm thinking about Legos/Duplos, trains with tracks that can be assembled and reassembled in different patterns, a dress-up corner (with old clothes of yours or things you find at yard sales or resale/thrift shops), puzzles or hopscotch squares to be assembled/designed, and marble game towers with tubes and slides to be combined and recombined. (You'll have to watch the marbles with a baby, but it's a great toy.)
You can also talk to her or have her playing on the floor while you are giving the baby a bath or feeding her (unless you are all eating together). She can help you read to the baby - which bolsters her reading skills by giving her simple books, and entertains the baby.
I know you have things you need to get done too - but sometimes you can make it a mother-daughter activity by letting her help set the table, sort socks, fold and put away underwear, etc. She will build skills and confidence, and get the reward of your praise and attention.
Art supplies may need to be in a special place so she isn't using them on the family room carpet, but a small organizer set (with a few drawers) in a corner of the kitchen would give her a supply of markers and crayons, paper and scissors, to use at the kitchen table. You can also have some activity books in there with mazes, simple word searches, etc. Things with glue and craft pieces can require some Mom supervision, but she doesn't need you to do it all with her every second. Go to the library and ask the children's librarian for some recommendations on books you can take out with some seasonal crafts or things a child can do on her own by looking at the step-by-step photos. Tell the librarian you are trying to encourage independent and creative play too.
Maybe every other weekend or even one day during the week, you can organize a play date. Start with having a friend come to your house (at a time when Dad is not sleeping), and later you can progress to her going to another child's house. You don't want to be running around constantly, I get that. But every other week should be more than manageable. Then you can include one outing per weekend, to the park or the grocery store or a children's museum.
I think if you have a very vague structure of "outside time" and "day trip/playground time" and "art time" and "play by yourself time", you can have enough flexibility for her and also time for the baby and getting laundry done and dinner cooked. If you need to relax your standards and not always vacuum or make beds, that's a gift you can give yourself and that's okay too.