Unfortunately that's just all wrong. The part you took from the second source isn't even about the rotating detonation engine but the detonation pulse jet engine. Maybe the exhaust velocities are the same but I doubt it. And the figures used aren't even the actual exhaust velocities. That's the speed of the Shockwave from the detonation and the speed of the wave from deflagration.

But rocket engines use something called a de laval nozzle. Designed for the flow to speed up to Mach 1 at the throat and then go supersonic out the back. So the exhaust velocity of a typical rocket engine is already in the several km/s range. For example, rs25 has an exhaust velocity of about 4 km/s. Twice that of the figure you used for the detonation engine.

You can't easily just take an exhaust velocity and calculate how long a trip to mars would take. The exhaust velocity is not a limit on how fast the rocket can go. It's more about showing its efficiency. Higher exhaust velocities are more efficient. This is also measured in a term called Isp, specific impulse. Which is why ion thrusters are so efficient. They cam have effective exhaust velocities of about 40 km/s.

With effective exhaust velocity (which I'm not sure 2km/s is it for an rde) you'd at least need the initial (or wet mass, aka fully fueled rocket mass) and final mass (dry mass, mass after burning all propellant) to get the total delta v from the rocket equation. That would give you a rough idea of where the rocket can get you. The more delta v the faster you can get somewhere.

That's not how rockets work. Vehicle speed changes as your run the engine and produce thrust (push). Earth and Mars already are in orbit around the Sun. To get to Mars, you have to change your orbit so the other end crosses Mars' orbit at the same time Mars is at that point.

2 years and 22 days? But the original quote said it takes 6 months to get to Mars at the moment?