Do the physics of this even hold up? Never mind how would you get the asteroid into orbit, then keep it in orbit, then keep airplanes and weather from hitting it.
The orbitology is a bit complicated.
Consider that as it's doing its figure-8, the building is going to be moving through the atmosphere at roughly the speed of a jetliner. The drag from that would need to be countered somehow. Not to mention the pieces that will get torn off, or the windows sucked out.
Because the orbit is inclined, the motion of this building and its tether -- which will of necessity contain a lot of conductive materials -- will generate a tremendous amount of electricity (see the Shuttle Tethered Satellite experiments), which also acts as drag. On the other hand, pumping electricity back through the building will act as propulsion, though to be of any use there'd have to be an enormous energy storage or generation capability.
And of course, there's the whole problem of building the thing. It would be built from the asteroid on down, which is probably not too bad for the parts above the sensible atmosphere, but those 600 mph winds down low will be a bit ... difficult.
Doing the rendezvous with the thing would be sporty. You'd have to rendezvous with the center of mass of the system, which will become more and more separated from the asteroid as the building took shape. (The asteroid would move radially outward, and the building would move radially inward.)
