I am creating a smart mirror by constructing a frame where a monitor sits right behind a mirror. I tried using a regular mirror but the mirror did not allow enough light to pass through to see the monitors lighting. Is there a material I can cover glass in to make it reflective like a mirror but still let light pass through?
What you are looking for is a so-called "one-way mirror" (Wikipedia). This is actually a partly transparent mirror: if the back is dark, it looks like a mirror but if the back is lighted sufficiently, that is seen through.
While I think are ways to make it work as you described, I think you will end up with either a poor quality mirror or a poor quality monitor - or both. There is a technical solution, though you have to consider technical cost and the fact that it isn't exactly the same as a mirror:
Use a Camera
Just a few years ago, this would have seemed crazy. But it is really easy to do now. Install a tiny camera (think "smartphone size") at the center top of the mirror. Make the default mode be: Echo the camera image to the monitor but with left/right flipped. This is what most smartphones do if you use the front-facing camera.
Then when you touch the screen or say "Mirror, mirror on the wall" or whatever triggers the "smart mirror", you can replace the "selfie" image with a menu or overlay it or whatever you want to do.
The only catch is that if you look at the mirror from any direction except "straight ahead", the image won't be quite the same as a true mirror. But unless the mirror is huge (wall sized instead of typical bathroom sink sized), that shouldn't be a big problem.