My treknobabble for FTL using impulse:
I always thought impulse engines are large gravity emitters internally powered by highly energetic reactions (possibly hybrid fission/fusion reactions) that burns fuel reserves. By themselves, the impulse engines are capable to drive a space ship to high sublight speeds. The power generated by the impulse engines (referred to as impulse power) can also used to power the ship's electro-plasma system (EPS) grid that powers most ship systems and equipment. One key system is the ship's inertial dampening system that generates a hull/skin tight anti-gravity field around the ship. This field results in three main effects: 1. a mass elimination effect; 2. external inertia and acceleration inside the field is mostly eliminated; and 3. a one g environment is maintained inside the ship. With mass elimination, the effects of mass gain, size reduction and time dilation are circumvented, so, the speed of light is no longer a speed limit and faster than light (FTL) speeds can be attained. Together, mass elimination and directional gravity thrust, you have the impulse drive. The maximum speed of the impulse drive is directly related to the peak energy levels supplied by the engines. Your fuel reserves are used up exponentially the faster you go.
The Enterprise has two impulse engines mounted on the rear of its saucer hull. Its engines are sized to provide maneuvering thrust around planets (orbit/deorbit, etc.) and emergency power and emergency propulsion in case of lost of its warp engines and their M/AM reactor systems. The speed and range for impulse power is limited by its fuel reserves and how hard you use the impulse drive. Since the Enterprise uses its warp drive and M/AM power systems as its primary source of power and propulsion, its impulse drive is of modest power and fuel reserves since it is only for planetary orbit operations and emergency uses. Totally guessing, I would put the maximum speed of the Enterprise on full impulse power at ~20c. As for fuel reserves, the upper speeds can only be maintain for a few hours (
TOS: The Doomsday Machine). At slower speeds like 1c or 2c, she can maintain it for a few days (
TOS: WNMHGB). Sitting still or drifting at a constant sublight speed with minor maneuvering adjustments, she might have power for a few months (
TOS: The Paradise Syndrome). I'm on the fence for the the SS Valiant in
WNMHGB, maybe she was a purpose built long range impulse ship, but she could be an early warp vessel, too, with a weaker impulse drive backup after all, it was 200 years ago. Her unexpected location on the edge of the Galaxy can be explained by the magnetic space storm. Form The Galileo Seven, we learn that magnetic potentials in space storms exponentially propel spaceships, just how much faster we don't know.
Ships build around impulse as their main power would have much bigger impulse engines/reactors (probably on pylons for safety) plus larger fuel reserves (flying gas cans) attaining higher speeds with longer ranges. The Romulan ship and perhaps the Eymorg ship, Balok's buoy/ship and the Doomsday Machine may be good examples of this design with the later ones using alternate power systems like ion power instead of fission and/or fusion reactors. Smaller cargo/transport/survey ships like the science probe vessel Antares, Mudd's class J cargo ship, the space hippie cruiser Aurora and standard Shuttlecraft (using an ion reactor

) may be other examples of FTL impulse drives. YMMV

.