Ditch Everest. Switch to Chimborazo in Ecuador. Two reasons: 1) Because it is closer to the equator, it's peak is actually farther from the center of the Earth. 2) Because it is closer to the equator, you get more of a boost from rotation of the Earth. Everest at 29.59 degrees North loses almost 14% of the rotational velocity.
So launching from Chimborazo gets you additional altitude, and about 223 km/hr extra launch velocity.
The structure you have envisioned is massively beyond our current tech. At both ends.
Digging a slanted tunnel roughly 100 km long is getting to silly proportions. The deepest mine right now is 4 km deep. This is getting pretty close to the limits of our tech right now.
The truss you envision holding the launch tube is grotesquely beyond what we can build now.
The launch energy needs to be supplied. You need electromagnets up the length of the tube. The energy they need to supply, assuming a constant acceleration, increases as the speed increases. So you need to run gargantuan power cables up the structure as well.
You might get someplace by forgetting the extension above and below the mountain. 234
You could have some sort of electromagnetic launcher that pushed a sled that carried your orbiter. Assuming a 50 km track and only 1g, you get 1 km/s in 100 seconds, pretty close to Mach 1, up the mountain. This is fast enough that the sled could detach and the orbiter take over its own burn. The first 25 km or so of the track would be level then curve up the mountain.
There are lots of variations on this. For example, the sled could also be a rocket motor that acted as a first stage. Or you could amp up the acceleration up the hill. At 4 g you get pretty close to Mach 2, in 25 seconds.
You can get an idea of what you are gaining from launching from the mountain. At 4g you are basically getting the first 25 seconds of rocket power from your launch sled. Suppose you were able to build the truss and extend the ramp another 50 km. This gives you only about another 10 seconds. The part on the mountain might be worth it. Building this currently-impossible truss seems to be a diminishing return. 234