How do I help my mom who is showing signs of depression or mild cognitive impairment?

Treatment. Help her to see her family doctor and accompany her to the appointment so the doctor knows your concerns. He/she can work to differentiate cognitive changes related to depression from those of early dementia (though some have both) your mother's physician can then make the appropriate referral for treatment if indicated.
A Checkup. Try and make an appt with her dr for a checkup. Often, lab tests can reveal some problems easily remedied, such as a low level of thyroid hormone. Sometimes medicines a person is receiving are the culprit and stopping them (under the doctor's supervision) can result in improvement. For example, certain blood pressure medications called beta blockers can cause depression and cognitive disturbances.
See her PCP. It may be difficult to have your mother go to a geriatrician or psychiatrist, but probably easier to get her to see her primary care provider who may already have a relationship with her. The pcp can then evaluate and provide the appropriate treatments. If your mother resists seeing any professional, it would be good to start by expressing your concerns about her and engaging her.