Take footage or an image of the blood pressure gauge without you in it and place it where that one is in the scene.
If it's a still, you'll need to add some noise to it to make it match the video.
You will then need to rotoscope the subject to keep him in front of it. You'll only need to work on the head (unless he gets up/dances around/flails his arms etc.).
Info on rotoscoping here: Rotoscoping introduction and resources
It's possible that the RotoBrush would work on this shot to save some time. Info on the RotoBrush here: Roto Brush in After Effects
If you've never used the RotoBrush before, follow the tutorials in that link first. It's not as intuitive as it looks.
For future shots, I would recommend putting the camera at an angle to the wall. Not only does that make the shot much more visually interesting by adding depth, but it also prevents reflection issues like this.
Here's what I would do. Stabilize the shot so that the Blood Pressure gauge does not move. Find a frame where the gauge face is the most fully exposed. Export that frame as a Photoshop File. Take the frame into Photoshop and minimize the reflection. You don't need to remove the whole thing because it's not going to be moving in the shot when you're done. Mask and crop the PSD file so only the face of the gauge is visible. Import that still into AE and then position it so it is right over the gauge. If there are any perspective changes to the gauge face then animate using corner pin so the face stays put. Now Duplicate the stabilized footage of scene and roto the actor. You only need to be careful with the edges when the actor passes in front of the gauge face. Then move the PSD layer between the rooted actor and the background layer. (This is better than making an animated mask on the PSD file because it will be easier to create believable edges) Re-introduce the camera motion to the shot and add the motion to the PSD file. Here's a short tutorial on how to do the stabilizing trick. The subject is different but the procedure is exactly the same.