No, the planet's orbital speed doesn't affect the escape velocity. That's just a function of the planet's mass and radius. However, it does affect the speed you need to escape from your stellar system.
There's a simple relationship between the speed of a circular orbit and escape velocity. The circular orbit speed is given by
$$v^2 = GM/r$$
where $G$ is the gravitational constant, $M$ is the mass, $r$ is the orbit radius. But I assume you already know that, since you correctly calculated your planet's orbital speed.
The escape velocity is given by
$$v^2 = 2GM/r$$
To calculate the escape speed from the planet, use the planet's radius for $r$. Of course, that ignores air resistance.
But once you've escaped from the planet, if you want to escape from your stellar system, you'll need ~878.3 km/s, relative to the black hole, which is $\sqrt2$ times the planet's orbital speed. That's not going to be easy!
To calculate the orbital speed and escape velocity close to the black hole, we really should use general relativity, but those numbers should be ok at 23 AU. FWIW, the Schwarzschild radius of your black hole is 29532.5 km.
By the way, we only know the value of $G$ to 4 or 5 significant figures. Instead, we normally use the gravitational parameter, which we can measure to much greater precision. That Wikipedia article has a table of values for our Solar System, but JPL has a better one. I have links & info here: https://astronomy.stackexchange.com/a/48616/16685