Knowing the Efficiency and Efficacy of Permanent Telehealth Options Act of 2021 or the KEEP Telehealth Options Act of 2021 This bill requires certain agencies to report on the availability and effects of expanded telehealth services under Medicare, Medicaid, and the Children's Health Insurance Program (CHIP) during the public health emergency relating to COVID-19. Among other information, the Department of Health and Human Services must detail its actions during the emergency to expand access to telehealth services and must analyze telehealth usage by certain demographic groups. The Medicare Payment Advisory Commission and the Medicaid and CHIP Payment and Access Commission must report on the efficiency and management of the telehealth expansion and related risks of fraudulent activity.