Clear the towers in front of your army with your archers and crossbowmen, build a siege tower with the engineers and use it to send your macemen on the wall and use them comined with knights to kill everyone in the castle.
Try to save as many ranged units as you can since they are your main line of defense against the two incoming waves. Also don't waste your knights since they are excellent for targeted attacks.
Improve stone and iron production to make gold by selling at the market. Exploit religion to keep higher taxes (ale is not necessary).
To prepare for the attack it is sufficient you make as many crossbows as you can (don't bother making leather armor, it can be bought by selling some of the crossbows) and many engineers to make shields to site on your towers protecting your archers and balistae. Make as many crossbowmen as possible (around 70 is plenty, with some 40-45 shields).
You don't have to scatter your ranged units on the towers if you don't want or are having trouble doing it quickly: you can simply place your entire army on the keep as long as there are enough shields to protect it and that should do it.