Use flying rocks as your siege machines
If there is one flying rock there are other flying rocks as well. Use them as a base for your siege.
As a general rule - get some flying rocks (they need to be large, but much smaller that the one utilised by the castle as a base), chain them to the ground/platform and use as a base for your attack. You can use them in various ways:
Make a posts for your archers that gives them enough protection and enables to shoot efficiently slightly downwards. Use a bit longer chain than the one used by the castle so the archer posts are over the castle and shoot from the above. You are less vulnerable to backfire from the castle since it's always easier to shot downward than upward
Balance a flying rock with regular rocks in a way that makes it easy pull the rock down with a significant force. Attach two long chains (longer than the one used by the castle) to the rock, with additional chain reaching ground attached to the holding chain. Place the holding chains on opposite sides of the castle. Now pull the additional chains with as large force as possible (possibly the chain has to go through some ring attached to the ground so that you pull parallel to the ground) to throw the rock down into the castle. It works as a huge hammer.
Use a similar appliance for a rock but balance it so that you can put some troops on top of it. Now when the rock is pulled down and hits the castle your troops can jump down and attack the castle defenders.
You can use various ways here:
- take flying rock as a base, fill it with stones and a small crew. Using chains control it so that the stone flies over the castle and drop stones you have. You can use other things, hot tar or boiling water for example.
- attach stones to small hot-air balloons. Launch them so that they fly over a castle. Now use archers (see above) to pierce the balloons dropping the stones on a castle
- similarly to the previous way but rather than using archers use some simple time trigger to pierce the balloon after a specific amount of time that ensures it's over the castle
Since you have access to the chain base, you can use it to detach the castle. Since you don't want it to drift away, use other chains to control the castle so that rather it floating away it goes where you want to.
You might need to do multiple re-chaining or something like that but it should be possible to move the castle anywhere you want.
Mountain as a siege machine
Pull the castle to the mountain, with a steep slope, that you can use to attack the castle.
Starve on the sea
Pull the castle to the sea and attach it to the sea bed. Then sail away and wait one year until defenders starve/run out of supplies. Later just come back and collect what's left.
You name it - freezing winter or a scorching desert Sun. Just get the castle somewhere were you don't want to stay for a long period of time
We may not be able to control dragons, but locating a nice supply of food (i.e. castle full of people and their edible supplies) near their nest should drag them for a feast. They will do the conquer work for you.
If something is up in the sky, it is vulnerable to smoke. Build large bonfires so that the smoke goes in the direction of the castle, add whatever you can to make it as smelly/poisonous as possible. You'll need to move your bonfires probably but just keep them in the smoke until the morale is so low they will beg you to allow them to surrender to you.
Cut oxygen supply
I don't know how high can a flying rock fly. But if you can reach over 10k feet with it you end up with an oxygen deprivation. The higher the better. At 20k feet your defenders will soon die due to the thin air.
Just make sure the additional chain you have to add is strong enough to pull the castle back to the ground once your defenders are done.