Followup question regarding the royal justice system (I’m assuming the steps you advise will be met with resistance from the lords across the realm). What would you offer these lords in return, in order for you to placate them?

I would offer them the right to nominate a certain number of candidates to the list of potential appointees, such that they had some say in the overall process but the king has the overall control, and suggest sotto voce that the king would not be averse if some supernumerary sons got cushy government jobs for life that got them out of the family house.