looking for a dual-monitor KVM solution w/ wireless keyboard+mouse

i’ve got an end user who has a standard computer monitor setup at their desk in their office, and another computer in a different room (sharing a wall with their office) w an HDMI cable routing that computer to a wall-mounted TV in their office. KVM switch is mounted to the wall underneath the TV. the user wants to be able to switch between these computers while using the same wireless keyboard and mouse. current KVM switch doesn’t work w/ a dell or a logitech dongle, and from what i’ve gathered people say logitech either definitely doesn’t work with KVMs or are great with them, lol.

i’m thinking the KVM switch needs to be replaced either way since i was told by the end user that it was a super cheap one, but i also want to make sure they get a wireless keyboard and mouse that will work with the switch. any reputable and reliable combinations?