Metals don’t properly bounce light with lumen. You can imitate the effect however. It will have some side effects and maybe not be exactly physically accurate.
You will need to use the “Ray tracing quality switch” node to disable the metallic value when evaluating the light bounce. This will make lumen treat the mirror as a white sheet when evaluating global illumination, which will create a bounce.
You will also need to disable screen traces, because you need to force the GI to be raytraced.
The reflected light will be too soft and diffuse. But it might be good enough.
Another option is to use a second light to fake the reflected light.
You can try to increase the resolution of Lumens voxels with console commands. It may make it sharper, but as I mentioned before this technique unfortunately has its drawbacks and one of those limitations is that the reflected light will be too soft.
i responded to another mirror case a couple days ago. lumen is a GI solution. it does soft ambient lighting. it does not have the resolution to create hard shadows, like a pathtracer. this is realtime, not offline rendering tech. if you need hard shadows you gotta cheat to create the visuals. it is what it is.