By one of those weird coincidences, I had been on JP Aerospace's site mere hours before reading this question about space dirigibles here. Their mission plan seems too good to be true.

They are proposing a three part infrastructure

Earth to the top of the atmosphere.
High Altitude airships fly from the ground to the station at 140,000 feet.

Transfer point at the edge of space.
A two mile wide station parked at 140,000 feet is the new way station to space. The station acts not only as a port for the orbital airship but also as a research center, construction site and tourist destination.

From the edge of space to orbit.
This 6,000 foot long vehicle never touches the ground. This airship flies from the upper atmospheric station to orbit. It uses hybrid chemical/electric propulsion to slowly accelerate and reach orbit.

L to R: Airship, Station and Orbitor (not to scale).

Set aside, for the moment, the engineering & technical challenges of establishing a permanent, crewed "suborbital space station" two miles wide at 140,000 feet. The company does, after all, hold the world's record for the highest flying airship. What really lifted my eyebrow was their Orbital Ascender:

This is a huge (a mile or more long) lighter-than-air craft to be assembled at 140,000 feet.

The airship uses buoyancy to climb to 200,000 feet. From there it uses hybrid electric/chemical propulsion to slowly accelerate. As it accelerate it dynamically climbs. In nearly nine hours it achieves orbital velocity.

They're a bit cagey about their "hybrid electric/chemical propulsion," but the ion engine (or whatever) is to be powered by solar cells overlaying most of the top surface.

Is there anything about high-altitude dynamics or orbital mechanics that renders the concept of an airship reaching LEO orbit unlikely? Or are there any other show-stoppers to this concept?

If you're interested, there's a book by the concept's originator:
Floating to Space: The Airship to Orbit Program, by John M. Powell (Apogee Books Space Series, 2008)