How do I make my mother feel like she is contributing?

My mother does not have major cognitive problems, but her mind is not what it used to be. However, she cooks, sometimes 3 times a week, and makes out a weekly menu and shopping list. She does her own laundry, and folds mine.
Mentally, she is always much sharper on days when she is active.

Try something simple, like folding laundry. That may also give her a sense of purpose. I've also asked Mom to straighten out disorganized drawers in the kitchen. You'll have to try to judge what activities she's going to be able to do without too much frustration. But I think the key is that it makes her feel as if she's contributing to the household.

Not much help but my mom said the exact same thing it broke my heart. She said she feels like she is useless and doesn't feel like she belongs. She also can't do much for herself. I just tell her that I still need her and can't imagine what I'd do if she wasn't still here to give me moral support.

Folding laundry, washing dishes, stuffing envelopes, following a recipe with your assistance, vacuuming, sweeping.
Any thing she is physically able to do, she can do with your patient guidance. So important for them to feel purposeful. Especially if they feel they are helping and contributing.
It is good that you care. May God bless!

How old is she? Is she capable of doing simple repetitive tasks? There are a lot of volunteer jobs that are not very difficult - in fact boredom is a problem when organising the jobs. Putting together sandwiches for Meals on Wheels, with all the ingredients in a line on the bench, could be one example. Weeding a nature strip is another. Listening to children read in primary school might even work - it depends on just how able your mother is to read 'Dick and Jane' herself. There are some things where it is a big advantage to have an adult present with an alarm to press if any help is needed. Depending on your mother, it might be good to think about simple 'work' rather than entertainment.