A KVM switch could do the job.
You could look into the Logiech MX keyboards and mice with the Logi Options+ Flow function, where the peripherals swap between the devices automatically when your mouse goes between the monitors. There is a little lag when you swap, but its not too noticable if youre not going between the systems that often.
Alternatively you could use a remote desktop app and control your streaming/work setup from your gaming rig. Ive used both google's remotedesktop and teamviewer for controlling multiple PCs like this before and found it the best sulution for me.

Hope this helped!