First, I find it hard to believe that those are scaled similarly. The Imperial Star Destroyers (both the first and second class) are 1,600 meters long. Essentially, they are about a mile long. You could run from one end to the other in less than 10 minutes.
The Death Star was the size of a small moon. The first Death Star was a rough sphere 120 kilometers in diameter. In other words, a Star Destroyer's length was 1.3% of the Death Star's diameter. A docking bay capable of holding a Star Destroyer would need to be enormous. Let's assume the Star Destroyer is a rectangular block 1600 meters long, 800 meters wide, and 800 meters high (which it isn't, it is significantly smaller than that) and that a docking bay needs to hold two of those with sufficient space for clearance and maintenance hardware. Let's say this results in a bay that is 1800 meters long, 2400 meters wide, and 1000 meters high. That is a volume of 4,320,000,000 cubic meters (4.32 cubic kilometers).
That sounds HUGE, right? It really isn't, compared to the internal volume of the Death Star. The Death Star has an 60,000 meter (60km) radius. Plugging that into the calculations for the volume of a sphere, we find that it has a volume of 904,779 cubic kilometers.
That's means the theoretical bay would be less than 0.0005% of the internal volume of the Death Star.
So you can see, there could be LOTS of places they could put a docking bay for a Star Destroyer.
That said, there's no real need for them to have a docking bay for ships that size.
The purposes of docking bays is to allow maintenance, restocking of provisions/fuel, personnel movement, etc for small craft. You park the Millennium Falcon, Chewie can climb up on top and mess about with hardware, you can attach feeder hoses to restock the water reserves, reclaim waste, refill consumables, etc. People can walk onboard or leave. All of this is needed with small ships, where space is as a premium.
Star Destroyers don't need to dock to accomplish this. All of their routine maintenance is done without docking. They were designed such that anything they can repair out of onboard capabilities can be done without needing to dock. They have personnel who are trained to work EVA in those rare cases where external repairs are needed, and they have machine shops and engineers who can produce most of the parts they could need.
Refueling is not something that has been addressed. We know (according to Wookieepedia) that the ships are powered by an 'I-a2b solar ionization reactor', which is referred to as seeming like a miniature sun. This reactor uses fully half of the volume of the ship, and the fuel it consumes is unknown. The Imperial Star Destroyer is listed as carrying 'consumables' for 2 years for its standard crew complement (according to a rulebook published in 2014 for a licensed game). Given that, we can assume that the ship was expected to not need to take on consumables for about a year at a time. It would be reasonable to assume that it refueled when it restocked.
Star Destroyers that aren't in need of severe maintenance would find it easy to orbit the Death Star (as it could any moon-sized object). Those which did need serious maintenance would likely have been sent to a dedicated repair facility, such as Kuat Drive yards (where many were constructed).
TL;DR: Your picture is way off scale, the Death Star is easily big enough to have docking bays big enough for Star Destroyers to land within, but it's unlikely that anyone would build them in because why would they need to land Star Destroyers within the Death Star?