Problem number one is that the light is coming in at rightangles to the surface, like sunset on Earth. It will be brighter than sunset of course because there's no thick atmosphere to traverse, but it'll still be horizontal rays and that means every object will cast a long shadow and if everyone wants to see a littl sunlight you'll need to ramp up the terrain from the sunward end to the far end.
Alternatively, you might be able to fit a big reflector into the space that ruins through the middle of the cylinder to reflect the incoming light evenly over the surface, but the problem you have then is that a) you've filled your sky with a load of stuff, reducing what you can do with your interior space and b) you're capturing light from a small area and spreading it over a large area, so it won't be as bright as you might like.
Solar irradiance at Earth's orbit is about 1366W/m2. You're collecting through an aperture with an area of about 50 million square metres, so that's 68.7GW. The total internal floor area of your habitat is ~654 million square metres, so the average irradiance will be about 105W/m2. That's relatively low... equivalent to average irradiance of the more northern bits of Europe, perhaps, but it isn't entirely gloomy. Note the use of the word average there, so it won't seem like a nice sunny mid-day as seen from Stockholm, but more of a gloomy overcast autumn day.
So the good news is that your solar intake isn't too bad, but it isn't very high and it won't be enough to generate a good facsimile of summer. You can fix this in various ways.
Adding an artificial light down the middle of the habitat driven by solar power is perhaps the most straightfoward option. You don't need to build any giant reflectors, and you get perfect control over the intensities and wavelengths of your light, and can easily dim or brighten the system to simulate seasonal changes. In this situation you can just make the endcaps opaque, which probably makes other aspects of the station's design and operation easier.
Alternatively, you can build light-collecting mirror arrays ("solettas") at either end of your habitat, and light-distributing mirror arrays inside your habitat. The area of these solettas can be much greater than the area of the endcaps, so you can collect almost arbitrary amounts of light this way. This allows you to replicate even equatorial desert insolation, if that's what you really wanted. Keeping the mirrors in good condition and operational, and keeping the internal transmitting and reflecting elements clean and everything aligned nicely will be more complex than the artificial light option, but it will work and if you're building a habitat on this scale anyway you should have more than adequate knowhow when it comes to space operations.
Here's a very quick and dirty diagram. I feel like it'd be better if I'd just scrawled it on paper, but no matter. Shown are two example light collecting arrays, and one internal reflector design, which will hopefully make the above text seem a bit clearer. I've omitted your asteroid for clarity.
