only to add to Shelley's commentary ... you might consider expanding the search geography such that you could use the Metropolitana for purpose of arriving to key center city locations.
It's not a big subway system (https://www.rome.net/metro43) but locations near stops might open up housing opportunities while not interfering too much with your idea of "on foot". I've stayed at hotels literally next to major attractions (Trevi, Sp. Steps, Pantheon), but never in a room with kitchenette. Sorry, can't give a specific recommendation, time for a little homework on the usual websites.
Enjoy Rome! No limit to the attractions! Consider if there is sufficient value in a couple guided events - maximize your experience in the limited time available.