How to handle flying Mt. Everest -- Att: Bruce Willis ;-)

 If you explode a fission device in the closed end of a 10 foot long cardboard tube, in about 20 nanoseconds the entire blast should have exited the open end.

Kosta Tsipis weapons tutorial: Bulletin of the Atomic Scientists, early 80s: 99% of the energy of fission explosion is released in the last 8 nanoseconds.  Photons, which are the main output of the device travel 10 feet in 10 nanoseconds.  Hence, about 20 nanoseconds.

Richard Rhodes' 1995 book Dark Sun, on first page behind first set of photos: diagram illustrating the successive stages of a thermonuclear explosion.  The force of the fission blast at one end reflects off the walls down to the opposite end where it compresses another plutonium device to critical mass.  Trapped between two fission explosions, tritium-deuterium atoms are heated fusion temperature -- whence they release left over protons from the t-d fusion into helium.

This fusion reaction would peter out to less than a kiloton -- if the whole device were not wrapped in reactor grade uranium which explodes upon being bombarded with the left over protons -- which latter reaction feeds back and forth, back and forth until the process plays itself out.

The idea is that cardboard or steel sides of a fission device cannot move 186,000 miles a second to get out of the way of a nuclear explosion.  You could contain a nuclear explosion for several nanoseconds by wrapping it in toilet paper.
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While watching the movie Armageddon it occurred to me that the NASA characters might be able to rely upon the cardboard tube model to nudge the asteroid away (not one the size of Texas or course [!]; just the standard dinosaur wipe out-6 mile wide model).

If a fission bomb embedded down a hole can release 99% of its energy without breaking up the asteroid the next question would be how much thrust will the fission exhaust alone create since it is not blowing out much of the asteroid mass.  Multiply how much of matter in the bomb will be turned to energy and by the speed of light, squared; and that will give us our thrust -- I guess.

Now, we just have to find somebody who knows what they are doing to answer that -- or tell us that the whole idea is bulloks.  ???  :-O

Noto bene: cannot use a hydrogen bomb for this purpose because the back and forth fusion/fission process is self contained -- and will get around to taking out part of the asteroid.